ICONE DI MATEMATICA

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Transcript

ICONE DI MATEMATICA
Anna Maria Fornara, Elena Angela Porta
Icone di matematica
Percorsi di recupero e approfondimento
1 A
soluzioni
ALGEBRA
ELEMENTI DI TEORIA DEGLI INSIEMI
EserciziI
2 a) Sì; b) No; c) Sì
3 a) Vuoto; b) Non vuoto; c) Vuoto

 ; c) T = trapezi,

parallelogrammi

 ;

4 Rappresentazione tabulare: a) A = −1,0,1,2,3,4 ; b) B =  0,1,2,3,4



d) Z = ariete, acquario 

 poligono di tre
 lati ; c) C = x ∈N
 : x= 2n+1, n =1,3,5,7,9

5 a) A = x ∈N : x = 2n, n = 1,2,3,4,5
; b) B = x :x è un
;

{ } {( )} ( ){ {} ( } )( {) } ( ) { } ( )
{ } ({ ) } ( )
{ } ( )
{ } ( ){ } ( )
{ } ( ){ } ( )
{ } ( )

 che inizia con vocale
d) D = {x : x }è un( nome
) femminile
{ } ( )  
6 a) Finito; b) Infinito; c) Infinito
{ { } } ( ( { ) ) } { ( } ){{({ {})}} (}(( () )) )
{ { } } ( ( ){{) }} ( ( ) )
 ; e) F  0 ∈P A ; f) V; g)
 V 
7 a) V; b) V; c) V; d) F  5 ∈A
1
  ⊄ A; Φ ⊂ A  
8 −1∈A, 3 ∉A, ∉A; −1,0,2 ⊆ A; 1,2,3
2
9 A e B sono uguali
)    2 3 = 8 elementi
{ { 10
} } (P ( A ) =) {φ,{ J}, {Q} , {K} , { J, Q} , {Q,K} , { J,K} ,{{ J,{Q,K} }}}; (P( A ) contiene
11 a) Sì; b) No; c) Sì; d) No
} ( { ) } { ( { } ) }({ ( {) } ) }( ( {) { ) } } ( ({ ) ) } ( )
{
{ { } (} () { ) { } }( () )
{ { } } ( {( ) {}) { ( } })( ({) {) } } ( {( ) {}) { ( } })( ( ) )
{ } { ( }) ( {) {} (} () {) {} (} () {) {} (} () {) {} (} () )
A5 = {5,15,25
) {} (}; A7() = {7,17,27
) {} (}; A8() = {8,18,28
) {} (} ; A9() = {9,19,29
) {} (} () ) 
{ } (}; A6() = {6,16,26
} ( )
{
  ; d) 15,19
  ; e) 1,3,4,6,7,8,9,10,11,13,15,16,17,18,19,20
   


13 a) 2,5,12,14,15,19 ; b)  2 ; c) 2,12,14
  (soluzione
 non
 unica)

14A = b,c,d,e,f ; B = c,e,g,h,i
 B    
16 a) A − C ∪
  C −  A ∪
 B ; b) A ∩ B ∪  C−  A ∪
 
 ; A2 = 2,12,22
  ; A3 = 3,13,23
  ; A4 = 4,14,24
  ;
 
17A0 = 0,10,20,30 ; A1 = 1,11,21
Sono 10 e costituiscono una partizione.
{{ }} ( ( {{ ) ) }} ((  ))
18A = 1,a ; B = 2,b,c 
{




verificai
 ferro,

 fermio,

1 Rappresentazione tabulare: a) A = 1,2,3,4,5,6 ; b) B = fluoro,
fosforo,
francio



2B = x ∈N : x = n 2 +1, n =1,2,3,4,5
{ } { ( }) ( ){ } ( )
{ } ( )
{ } ( )
{ } ( )
 }{; e)( V;} f)){{F({  {7
3 a) {V;{b) }V;}c)( V;( d)) F) {11∈S
})}}∈P(}(( S{());) g)){}V; h)(}V;{i))({F }Φ) }∈P( ( S );) j) V; k) V;
 }},{11,13
{( } )}sono
)}( ( una
)     
( )l) F {{5,7,9
( ) )partizione
} {(( }){)},{(15


4A = B
5 a) Finito, non vuoto; b) Infinito, non vuoto; c) Infinito, non vuoto; d) Infinito, non vuoto; e) Vuoto; f) Finito, non vuoto
6 a) Non disgiunti; b) Disgiunti
(
)
(
)
( )
( )
c)  A − (B ∪ C )  ∪ B − ( A ∪ C )  ∪ C − ( A ∪ B ) 

 
 

 { di 2}; A (∩ B{)= {1}; B}(∪ C( )= N;) A ∪C{= {1} ∪}({numeri
9A ∪ B = {numeri
( ) potenze
( ) }{ )pari
( }; ) (  )
} dispari,
 } dispari
 }{ pari
A ∩ C = {potenze
( } ) (  )   
} { di( 2}{)− {(1}; N)}(− C(B=) C)= {numeri
{( )}; B( = CC)= {numeri
{{ 10
}}{{ ((1; }1}){){, ((1; }2}){){,((2; }1}){){,((2; }2}){){,((3; }1}){){,((3; }2})),((3; 3));A= {1,2,3
{ } } ( ( ) )    
11A1 = {Alessandria,
) };A2( = {Torino,
) } Biella,
( )Cuneo
) } Verbania,
( ) Vercelli
} ( { Asti
{  }; A 3( = {Novara,
{  } ( )  
8 a)  A ∪ B ∩ C  − A ∩ B ∩ C
b)  A − B ∪ C  ∪ C − A ∪ B 

 

13 Si possono ordinare 18 menù differenti.
14 9 studenti sono ambidestri.
verso il triennio e oltrei
1 a) B; b) E; c) E; d) nessun diagramma; e) C; f) D; g) E; h) C; i) B; j) A; k) C; l) E; m) E; n) B; o) D; p) B; q) B
2 6 bandiere.
3 Il numero di palline ruvide rosse è uguale al numero di palline lisce verdi.
2
 





ALGEBRA
INSIEME N E INSIEME Qa
436
7 0
EserciziI
80
912
105
11120
1210
13 a) M.C.D. = 24 m.c.m. = 5040; b) M.C.D. = 1 m.c.m. = 22950; c) M.C.D. = 1 m.c.m. = 6072
Per approfondireI
1 3 giorni
2 2 ore e 30 minuti
6 36
5 15
EserciziI
1 13
16
2 12
1 138 1316 1
17
4 27 129 4
soluzioni
CompletaI
2512 35
3 925 palline
38 16 1 13
18
7
9 2 12
11 13
1 38
38 16
19
24 12
7 49 7
16
208
9
1 13 1 113 38 1 1638 16
21
22
2 12 2 412 7 4 9 7
9
verificai
 F
( {{;c)) F}} 13(( 2{; d))) }F 2{(35{; e)}) F} (150
( ); )f){F {1}; g)} V;( h)( )V;{ )i) F }3{; (j) V;} k)){ (F{impossibile
{ { } } (1
{({ ) a))}}F (5({5 ;)b)) }F48
}) } ( ( ); )l)F 1; m)
2 a) Commutativa dell’addizione; b) Associativa della moltiplicazione; c) Invariantiva della divisione; d) Associativa dell’addizione; e) Invariantiva della sottrazione; f) Distributiva della moltiplicazione rispetto all’addizione; g) Distributiva della
moltiplicazione rispetto all’addizione; h) Distributiva della moltiplicazione rispetto all’addizione; i) Distributiva della moltiplicazione rispetto all’addizione; j) Distributiva della divisione rispetto all’addizione
3 a) M.C.D. = 9 m.c.m. = 23814; b) M.C.D. = 36 m.c.m. = 432; c) M.C.D. = 19 m.c.m. = 114; d) M.C.D. = 1 m.c.m. = 39270
4 9; 14
5 360 s6 a = 3 e b = 14 oppure a = 3 e b = 42 ecc. La soluzione non è unica.
7 e; d; a; b; c
8 108
9 0
verso il triennio e oltrei
112100, 4200 , 3400
22
 1
2  
 2
n
10 1
22
11 1
120
13
16
25
n
 1
3
a) 13; b) 590; c) 754b 5c
 2 
 1
 1
6 59,85 euro
7   con n = 1, 2, 3, 4, 5, 6
 2 
 2
Insieme Z e insieme Q
EserciziI
29
3 −16
625
8 −1
96
64 4
1
47 8 13 289
1
170 3
159
16512
−
−
81 15 81
2 3 7 144
4
8
81 18
64 4
1 64 47 4 8 113 47
2898 13
1 289
170
5 318 125
64 431
13 3289
1
1 125
17047
5 151 1 31 17015 31 125 5
−
19
− 18
−20 −11
−
−
− − − −− − −
−
−
−
−21
−
81 15 81 81 2 15 3 817 2144
43 8
81
18
12183 3 27 412
81
144
4 143 8 20
3
7 144
4 15
8 812 812
4 8114 18 202 12
435
10Impossibile 1116
171
5 64
126
132
14 −
Per approfondireI
64
64 64
4 4 141 47
147 847
8 13
8 289
13
13
289 289
1 1 1170
170 170
31
31 25
31
25 25
5 5 51 1 3
13 15
315 151 1 1
1 a) ; b)
2 − − −− − −
−− −
− − − − − − −− − −
4
81
81 15
81
15 81
15
81 81
2 2 323 737 144
7
144
4 4 144
8 8 881
81 18
81
18 18
2 2 12
2
12 12
33 4
3 4 14
414 14
20
20 20
verificai
1 15 31
170 1
64 4
1
47 8 13 289
64 464
471 31
8 4713
1 1 4170
25 8 289
513 289
11 3 170
2531 525 15
31 15
3
1
264
47
− 1
− 3 −− −
−− 5
− −− − − − 6
− 71
−− −−
4 1581
4 144
81 20218 122 12
81 15 81
2 3 7 144
81
1581281 18
3 2 72 3 144
8 81
127
384 4 8811418
3
43 14
4
1
170
31
8 289
13 289
1
170 2531 5 25 15 3 1 153
15
1
1
115; −5
− 8
− d, −a,−c −− 10 −
−
−−
− 9− b,
47 144
84
81
1881 218 12 2 312 4 3 144
3 144
8
20
14
20
12
 2  2
 − 3  ⋅  − 3 
 2
 − 3 
7
−
15
1
14
20
5

5
3
 2
 3
: −
= − 
 2
  3 
⋅  − 23 
−1
verso il triennio e oltrei
1 Ordine di grandezza: a) 101; b) 10–2; c) 106; d) 10–2; e) 104; f) 10–4
2 a) 0,00007; b) 1700000; c) 0,0045; d) 50000000
3 a)
11 1 77 7 99 9
; b)− − − ;−c)− −
6 6 6 6 6 6 252525
434 ⋅ 2325c
ELEMENTI DI LOGICA
EserciziI
2 a) Sì – F; b) No; c) Sì – V; d) No
3
−
1
20
A
4 a) «Elena dipinge», «Elena non guarda la televisione» collegate da “e”; b) «Finisco di leggere», «vado al parco» collegate da
“se … allora”; c) «Marte è un pianeta», «il gatto non è un felino» collegate da “o”; d) «Luigi cancella la lavagna», «Marta
cancella la lavagna» collegate da “o”; e) «Firenze è nel Lazio», «14 è un quadrato perfetto» collegate da “se … allora”;
f) «Un numero è primo», «un numero è divisibile per 1 e per se stesso» collegate da “se e solo se”
((
())
) ((
( ))
)
 rrV;∨∨c)ppr→
→
→ppp
→
q↔
→q
pq∨∨ rr q ∨–
∨ qq
p→ q – V
8 a) pp ∧∧ qqp– ∧F;qqb)
9a) «O non guardo la televisione o vado alla festa». F; b) «Se guardo la televisione allora non vado alla festa». V; c) «Non è vero
che non guardo la televisione o non vado alla festa». F (oppure «Guardo la televisione e vado alla festa»); d) «Non guardo la
televisione se e solo se vado alla festa». V
10 Vera
11 a) V – V – V – F; b) F – V – V – F; c) F – V – F – F; d) V – V – V – F – F – F – F – F
12 a) Generica; b) Tautologia; c) Contraddizione; d) Generica
verificai
1a) Sì; b) Sì – F; c) Sì – V; d) Sì – F; e) No; f) No; g) Sì – V; h) No
( ( ) () ( ) )
r qb)∨ pq ∨
↔p r ↔ r – F
2 a) q → q p→∧ rp –∧ V;
3 a) «Luisa è interista o Claudia non è juventina». F; b) «Se Luisa è interista allora Claudia non è juventina o Mariarosa
è milanista». V; c) «Se Luisa è interista allora Mariarosa è milanista, o Claudia non è juventina». V; d) «Luisa non è interista
se e solo se o Claudia è juventina o Mariarosa è milanista». F4 a) V; b) V
5 a) V – V – V – F; b) V – V – V – V; c) V – F – F – F; d) F – V – V – V – V – V – V – V
verso il triennio e oltrei
1 Si devono girare due carte, quella con il gatto e quella con il 3.
2 Elena deve aprire il pacco con il fiocco dorato.
RELAZIONI E FUNZIONI
EserciziI
1 Rappresentazione tabulare: R = 1; 1 , 5; 25 , 6; 36
{( ) ( ) ( )} {( x; y ) ∈ A x B : y = x2 }
Rappresentazione
, (5;caratteristica:
25) , (6; 36)}R = {( x; y ) ∈ A x B : y = x2 } oppure xRy ⇔ x = y con x ∈A e y ∈B
{(1; 1)per
2
{{ }}{{{((2; }6}}){), {((2;( }8)}),){{( 2;(10})}{){,((3; }6}){){, ((3; }9}){){,((4; }8}){),{((5; }5)}),{{( 5;( 10})}{){, ((6; }6})), ({(7; 7))} ( ) 
4 Rappresentazione tabulare:
{{ R}}={{ ((2; }5})){,{ ((5; }2})){,{((3; }4})){,{((4; }3})){,{((2; }6})){,{((6; }2})){,{((3; }5})){,{((5; }3})){,{((4; }4})){,{((5; }5})),((6; 6)), 
{{ }}{{ ((3; }6}){){, ((6; }3}){){,((4; }5}){){,((5; }4}){){,((4; }6}){){,((6; }4}){){,((6; }5})),({(5; 6))} ( )  
 x + y ≥ 7 con x, y ∈A. Proprietà simmetrica.
Rappresentazione per caratteristica:
( )xRy ⇔
{ R} = {( x; y )}∈A(x A :)x +{y ≥ 7} oppure
2 Rappresentazione tabulare: R =
5 Proprietà riflessiva, simmetrica e transitiva.
{
( ) riflessiva,

 antisimmetrica
} { {( a; }a{}) , ( b;( }b{) ,)( c;}c{), ( d; }d{) , ( d; }a{) ,( d; }b) ,{( d; c )}. Proprietà
6Rappresentazione tabulare: R =
e transitiva.
7Proprietà riflessiva e simmetrica.
9a) Antisimmetrica; b) riflessiva, antisimmetrica e transitiva; c) riflessiva; d) riflessiva, antisimmetrica e transitiva; e) riflessiva,
simmetrica e transitiva; f) riflessiva, simmetrica e transitiva; g) antiriflessiva, antisimmetrica e transitiva
10 a) Relazione generica; b) relazione d’ordine largo parziale; c) relazione generica; d) relazione d’ordine largo totale;


e) relazione d’equivalenza, classi ; f) relazione d’equivalenza,
classi di strumenti ; g) relazione d’ordine
stretto totale
11 L’insieme quoziente è Q =  3, 2, 6   
 

{{ }} (( ))
{ } ( {) } ( )
{ { {} {} {(} {(} )(} )(} )({ )( ) }) {( }) ( )
} ( f (7) non
  7 ∉D; la controimmagine di 3 è 2;
14D = {1,2,4,5
)  esiste,
{ }{ { (};}C}(=)( f ( A) ) =){0, 3,{}15,
{{ (}24}});({f({(2 ) }=))}3; (f{( 4{ )=)}15;




la controimmagine di 6 non {esiste,
6
∉f
A
{ }} ( ( ) )    
3
1
{1
 {))= }–} ;( f( 2 ) non
  2 ∉D
15D ={ Q
)  esiste,
{{− }{2}};( f(}(0{{) (=) )−}}23); f((–−4
2
22
 
di h è 2; la controimmagine di c non esiste,
16D = {2,3,4,5,6
}{ ( }; )C (= {b,d,e,f,h
{) {{} } }(}{;( f{(()4 )}=)})e;( f(6) =) f; la controimmagine
c ∉D; la funzione è iniettiva, non è suriettiva, non è biunivoca
{ {} ( } )( { ) } {( {{ }) }}({ {( ( ) } )}) ({ ({ ) }) } ( ( ) )
  4 ;f w
 = 3 ; la controimmagine

17D = k , t , u, v , w ; C = 1 , 2, 3 , 4 , 5 ; f t  = 2 ; f  v =
di 1 è u; la controimmagine
di 5 è k; la controimmagine di 2 è t; la funzione è iniettiva, suriettiva e biunivoca; l’inversa di f è ancora una funzione
 1  5  1 1
 11 55   11 15
1  1 1
 f 1 = 0;
 f++−4
 =  ; f +  = non esiste,
∉D
18D = Q − +  ; f 0 =
 1;
2  2  2
 2  9  2 2
 22 99  22 29
verificai
 3 , 3;
 5 , 4;
 7 , 5;
 9  
1 Rappresentazione tabulare: R = 1; 1 , 2;
{ { } } ({ ({ )})} ( {({ ) )}} (( ))
{{ }}{{{(( }}}){){ ((( }}){){) (( }}){){ (( }})) ({( ))} ( )  
( )xRy⇔ y = 2x − 1 con x ∈A e y ∈B
{ } {( x; y}) ∈A( x B): y = 2x{ − 1} oppure
Rappresentazione per caratteristica: R =
3 Antiriflessiva e simmetrica.
4
4 Antisimmetrica, transitiva.


ALGEBRA
6 a) Riflessiva, simmetrica, transitiva; relazione d’equivalenza. b) Riflessiva, antisimmetrica, transitiva; relazione d’ordine
largo parziale. c) Antiriflessiva, antisimmetrica; relazione generica.
7 a) Riflessiva, antisimmetrica, transitiva; relazione d’ordine largo parziale. b) Riflessiva, simmetrica; relazione generica.
8 Riflessiva, antisimmetrica, transitiva; relazione d’ordine largo totale.
{ {} (} () {{) }} ( ( ) )
{ {} (} () { ) { } }( ( ) )
 
10Funzione;
di 2 sono b e c; la controimmagine di 5 non esiste,
{ { } } ( f( A ) =) {1,2,{ }{3,{ 4,}( 6}}{(; )f{( (a )}=)}1;)(f(b) =)2; le controimmagini
{ { }5}∉( f( A ) )    
 b, c ; b)Relazione;
  c) Relazione; d) Funzione, D = 1, 2, 3, 4 , C = a,
 b, d,
 e 
9 a) Funzione, D = 1, 2, 3, 4 , C = a,
soluzioni
5 Riflessiva, antisimmetrica, transitiva; relazione d’ordine largo parziale.
 

11 a) Non iniettiva, non suriettiva; b) suriettiva, non iniettiva; c) iniettiva, non suriettiva; d) iniettiva e suriettiva quindi biunivoca
   1 1  1
 1  13   3 1  111  311  11
5 111 111 513 11 111   31 1 11
0;
12 a) Q; b) Q; c) Q − 0 ; d) Q −  ; e)
 –esiste,
 f  –  non
 ––Q−–+  ; f0– =−1;
Q−
  – 13
 2   – 2 ∉D3  2 
2+–f2212 =934 ;f22–=222
3
4
2
2


2
9
2
3
2
3
2







4
2















 2  24   4 2  222
   

{ { } }( ( ) )
{ { } } ({ ({ ) }) } ( ( ) )
verso il triennio e oltrei
1 Sì, le 14.00 del 15 ottobre più 24 h danno le 14.00 del 16 ottobre.
{ { } }2( f (3) = )9
  
3 La controimmagine di 27 è 6.
4 La controimmagine di 5 è 16.
5 “Uno a uno” è una funzione biunivoca; “molti a uno” è la funzione costante.
Monomi
EserciziI
111
222
111
33 3
855 5
5 553 5
2 a) 15a
15
15
15
aababbc;
bccb)c −− − xx33xyy333yzz344z; 4c) aa44abb444bcc4;c d) xx33xtt838w
tw
w
666
555
28
28
28
CompletaI
1
4
at
4 a2 + 2ab − 10b2 a2 + 2ab − 10b25 as +
3
21
EserciziI
13 5 7 13 15 7713 1 17 72 21 51 722 2 51 22 2 5 2
2 5
y −−4
x ybx9
−−− 8bxbcax− − x ybx −ax
− xy
6 x2 + 4y − 4
−xy
x2 + 4 y − 4x7 + 4 ac
x2 +bc
4 yac
ac
x y − xy
−−−4 bcax
36
8
6 8 9 6 28 986 2 69 8 2 36 8
3
CompletaI
7 2 2 3
1 27 3 2 2 3
1
xyzx yz 11 − a2 b3 xyz
10 − ab x yz − −a b ab
4
6 4
6
EserciziI
2
5
 2012
51
5 31556 7 4355 8103520
1
1 36 841
4 1136 
5 3 5
3 8 5
53
13
545 11036
516 26 6451210
7 8102
2 66
a b cx x y − − as36bt5713
cx abx 8 y−532a10
−a xbs cx
t −− xsab
ay14
tb5 3
cx2
−a10sx6xa5t876yx56−y12ab
− s s63
t t20
2
aa b xx5ab
ta −x3−2
y as10xt520x 36
a−yb72
xas10
tx2t6±
y −x 3 y 2 az 64axx836
15
12 −
5
5
5
5
162
16 2
4
6425 
16
4
64
6
264
63 2 25
664
 3 4096
3416
3 409625 64
3
5 3 5 5 33 85 5 3 181065 57 5
1 6 7 5 10 5
1 36
45 510 2045 10
1 20
1 636
10
6 6 12
6 12
8 2 36
6 2 8 2 612
36 72
−
a
b
−
cx
a
x
b
cx
y
−
x
s
y
t
−
ab
s
t
3
2
a
ab
x
3
2
−
a
x
s
t
−
s
t
a
x
y
a
x
y
a
b
x
t
a
b
−
x
t
x
−
y
x ±
160
18 32a x 19
20
4096
16
16 4
4 2
6
2
6
4096 
64 25
25
35 64
35
2
2
3
2 2
1 4 3 2724   4 3 23 47 2
 2 n 3n −44 2
58 53 5 1 36 78 55
1 6 710 55
45 10 205
145
1 68 62 12
36 1
36
2 72
12
ya b cx
− s tx y ab− 211
3
s 2ta x ab− 32sa10
tx
− a6sx106 ty20
aa xb yx6t 2 − a8 b2 xx636
t23
y −  ± xx36
yy z ± ±0x , 4yx zy 
±0,1aab
bc 
±±00,,41axn3byn7− 4
22


6
2 4
6
2
6
6435
64
25 4096
25
4096
 3
 3

3

35
(
(
) ((
))(
)
(4−x 2− 2at ax x ) (4−(x452xx tyt )94) 94− 52−x 52xy xy y4p94 94
) (( (
)) ((
) )  − 32 3aba abxc c 
2


 21
    11

44
2
2
4 922
44 99
±±0x, 4
x yx y zyz )  ( ±±(0±0,0,14,a4xbx 25
y y ) ) a)( ±(±00,1,ab
a1abb
c  ) ;) b)Impossibile;
− ab
aabxc c   c) (4−x− aat x x   ); d)( 4(−4x x xt ty ) ) −− x x xy y y 4p
x x y y 44pp
33
3
5
9 455
99 44
3


 32
    22

   1
 2 4
1
2
4 9
y a x  ± ab(x4cxy z t   − ()27
a, 4x−x  yx )y ( 4
t
x y
±0
(x±940,1ta94bx )y ) −452p 23x aby28c 94 94 −x 21y a x4p (4 x 29Impossibile
) − 52 x y 30
2)
5
9 4
   2
 3 3
2

33623672
4 72
2 2
 4 43 32 724 24 
32 2
33 3
 2 53 4344 
1
n3 nn4−n4− 4   2
 141

2
1
61 18 62666612
3636
26 6
a abaxxxyt 2y12 − a8ab82xbx36
xty272
t 2 a)
− − ± xx yx yz y  ; b)±±Impossibile;
±0,x4 xy yz z c) ±±00,±1,a04,x4
b xy y ; d)±
±,0
1
ab
a,1abc b
24
 0
5 64
64
24096
525
4096
3
3
 4096
  33
 

222
33 3
22 4
74
3
2
4725 4
22
n 33
n −747
322
n3
n n4
−
n−
44
3 2
3 2
3
33 4n2− 24 3n − 4
5 43 72 3
33
3
b q =
33
n−
n2
− 23n33−
n−
44
3
3 5 5 4 3t
3t 2 33 4
Per approfondireI
ak = 1 h = 3
333
n − 2553n4−44
5334 4
3 2
n − 2 3n − 4
n3− 45
22 2
n 3n −374 7
3
r
2
s=3
n − 2 3n − 4
3 225 33
3
3t
3 3
n1− 2153n54− 44 3
33 55
2 3
3 3
n −n22− 2
33
n3−n4− 4
3 25 23 3 3t
3t3t
3 5
3t
4p314p3t
w =3
CompletaI
9
32
64
EserciziI
64 2
c 33 −
243
3
347a2
3 5 4
350
36 a) M.C.D. y , m.c.m. 12x y z ; b) M.C.D. b , m.c.m. a b ; c) M.C.D. y2 z2, m.c.m. 14x3y4z5
3715b3c3x soluzione non unica
2
3 3
387xy3z5 soluzione non unica
verificai
11
11 22 2 11122 2 111 22 2 22 22222 2 22 2 2
11
1 a) −−− ax
; 2d)
axax; b) yy y; c)
−−− ab
abab 2
x2
x xy2x
y y y−;−2e)
−b2
b−2b
bzz z z
2
333
333
888
5
) ((( ))((
))
 3 33 3   
3 a)   xzxzxz
; b)( 0(,05(,0a5(,0
xz
a5b,a5bcabc)bc )c);(c)
−
)(2−(y2−(2yc−y)2cy)c )c(0);(,d)08(,0x8(,08x,xy8 xyzyz)yz)z) )
 4 44 4   
 4

2 a)  ± a2 b6 c 4 
 5

A
2
3 33 3
4 44 4
4
17 2
a b 4a 4
3
(
2 2
2 22
2
  4 42 26 364 24 
; b)
Impossibile;
±±
±0a, 9abxybc c c)±0±,0
1±x, 16
9
1x16
y 8yz84z 4
0,xy
9y 83xyz34; d)±0±,10x,16
  5 5
 
3 33 3
2 2323423434 4
65 4 2
54a
−
pt
16
3 33 3
5 55 5
2
3 33 3
121212
6 12
636363 3
5 6 6
17
17 65
5 65 62
5
3 4
x 6Impossibile
y 14 x 3 y 3 4aa32bc
b 4 4a47 −a2 b pt42a4 − x6pt
y 8 14 x 3 xy63 y 6 4a14
bcx 3 y 3
144
3
3 16
14416
144
4a3 bc 4
9 a) M.C.D. q, m.c.m. p4q4r3t2; b) M.C.D. 3, m.c.m. 24a5b4c6x2y4; c) M.C.D. x, m.c.m. x2y6t2
1014x3y3 soluzione non unica
114a3bc4
verso il triennio e oltrei
2
1 2100 − 75π u (
)
 π 2
 3 9  3π2  3π2  2  π 51251
 15
3

315
 159 9 15
 32 93  2 3

51
51
b+π 3
+x 2p
a2 ba+
b π a b
ab+=π π
π+ πxπ xa; +
ab ab
2 Fig.
2ab2p = + +
Fig.
= + ab
;ab
Fig.
 44
 2 +1π2p
 x =  2 + 3  ab; Fig.
 +2p
 ab
4 8 8 4 8 
2
2
24 2 8  22 23  3  2 23 2
3 a) n, p ∈N, n ≥ 1, p ≥ 3; b) k, t interi, k ≥ 0 multiplo di 3, t ≥ −5; c) n = 4, p = 8; d) k, t interi, t ≥ 1, k ≤ 0
4 a) 2–1xyz; b) 3132–5a14x–17; c) 2–33–9x15–15ty18–3k; No, non sono espressioni tra monomi.
Polinomi
EserciziI
2 a) 6 – 5,2 – No – No – Sì – Sì – No; b) 8 – 6,4 – No – No – Sì – No – Sì; c) 4 – 4,4 – Sì – Sì – Si – Sì – Sì
33 3
11 1 22 2
33 3 11 1
11 1
33 3
3 a) −−aa−66a−−6 − aa55a++55+
yy22y 2 ; yy33y++3 +
5aa544a++4 + aa33a++3 + ; b)−−xx−33x−−3 3−3xx322xyy2+y+ + xx +x+ + yy33y++3 +
yy22y−−2 3−3xx322xyy2+y+ + xx −x− x−x33x 3
77 7
44 4 33 3
44 4 33 3
33 3
44 4
4 x5 + x4 + 1
5 t3 + 5 + t + t2
6 p4 + p3 + p2 + p + 1
7 a2b + 1 + b2
33
3
1
1
3 2 23 2 112 2 332
3
11
3
5 2 4 242 3 3 2 22
2 2 4 2 23
+ tx+5 +t 2x 4 +p14 + tp33++5p+2 t+
x84a
+p+t+x1 +p1a +
++1+5+p+b t++pt4
+a1 −p10
a +bbp+9
1++pb22b+2p−+41
a −a−10bbt++1+ b b −−410
aab−+10
−b3b2t + b2 −− ab +− b3 t + 11 − ab + b3
−bt p10b
22
22
2
6
3
2
2
63
2
63
2
CompletaI
1
7 2 3 4 3
2
5 6 12 2 5 47 2 3
3
a b +x a b − a4 b13
2ab5 x6 − a2 b5 x 4
12 − abxz + a b − a b 2ab x −− abxz
3
4
33
4
3
EserciziI
7
17 3 220
6
3
2
4
62
3
2
4
2
4
2
14 x5 + x 4 − x 3 y − a7 b4 + 2a5 b5 + xa55+b4 x 4 −5abx2315
+y 9b3−x2 −a73bax
+−23a5 b5−+22ba35xb24+ 9ax
−5+ab
14
+ 9b−37x2ab−23+ax −b33x2 + −22ax
b x+ + 9a
3
3
3
5
5
5
3
53
5
5
3
3
7 3 2 17
20
29 2 35
4 7 4
2 2
3 32 2
33 22
5 5 2
5
4
2
2
2
2
14; c) −7ab + b x +
y − a b + 2a b + 16
a b a) −5ab
−5ab+ +9b
9bx x − −3ax
3ax−−3;3 b) −22b
−22b xx ++ 9ax
9ax + 14
ax +
5 x + 3 y + 6ab −4 y + 3ab −
x −
3
3
3
2
2
3
3
29 2 35 2 23
2 2 2 2
22
2 7 3 2 17 1720
−7ab + b x +
ax + a) 5x 5+x 3y
+ 3+
y 6ab;
+ 6abb) −4y
−4 y ++3ab;
3ab c) −
x −
y +
ab
3
3
3
2
2
2
CompletaI
3
227
185a3x − 6a4 − a2x2
5a3 x − 6a4 − a2 x219 x2 −
abx + a2 b2
2
60
EserciziI
20 −4a2 − b2
2110x2y + x3y2 + x3y3
22 −2x3 + 9x2 − 20x + 8 23 −6x2 + 16x − 2
0xx + 8
−6 x2 + 16 x − 2
24
27 3 27 2
277 2 11 3
x +
x y+
xy −
y
8
8
24
2
CompletaI
(
) ((
)) (
)
(
) (
)
22
21
2
2
1
1 2
1
12
1
4s+2t 2 −b26
2st +4s2t 2 − 2Ast−+B = BA− −AB9a2=
+B
3ab
− come
A+ b2il precedente
4s2t 2 − 2stessendo
+
A−B = B− A
25 9a2 + 3ab + 9a2b2+3ab
27
È
4
4
4
4
4
4
EserciziI
284x2 + 4x + 1
2916 − 16a + 4a2
301 − 6xy + 9x2y2
31 a2 + 4ab + 4b2
9
9 1 2
9 12 2
1
1 22 4 1 232 3 2 44 12 23 3 2 22 42 2 3 2324 12 224 3 23 9 241122 22 4 92 11 26 2
2
2 2 2 2 2
34
35
+ 9x
1
y4+a29
16
x2−y116
− 6a xy
a+24++a94
xaby+1b−32
6axy4
++−
49ab
6x2x+y+2b2 xa224+−44
6ab
x−+a b+2x233
4
a −46−x a2
+5
+a2x−
a5ab4+−2a5b+a22 − 5axab
y+ −26b5xa y− 5+
xab
9yx+−y6bx y a+xb9 yxc y−−
6abcy
x xyay++b−9cyx6x−yyabcy
+a+ yb yca +− abcy
a ++ ya +b
4
4
4 16
416
4 16
4
4
16 43
8116 43
8
1 22 4 2 2 43 3 1
1
1
2
2
4
4
1
1
9
1
9
1
4
1
4
9
1
4
1
2
4
4 2 2
2 2
4 4 26
62 2 66 42 44 4 42 446 4 2 22 6 2 2
bx25
y5a−x −6y5x ab
−y 6++x 39ybx324+y2936
xx42yy2a42−b26cxa232−yb32abcy
c+29−x+abcy
y y+ ay22b37
a++ +a a+2y 2 a2a6 b2 −+
a62ba2a4−b+38
2+a aaba42bb+
a− ba2abba− +
2
baa+abbb+a+ab b+a 39
b + xa24nb−42
x+2xnn−+a122+xbx2n 2++1+ xa24k x+2a2n4a−k32
+k x+2naa+321kk++x
c +− abcy
4
4
4
4 3
3 9 49
4
4 16 16
3
3814 81 16 3
81
3
9
40 x2n − 2xn+1 + x2
CompletaI
42 a2x2 − 1
a2 x2 − 143
EserciziI
46 x2 − 9
6
4 2
t − 1 1− x 4 y 2
81
474 − 9a2
2
5016a − x x − 9
2
41 a4k + 2a3k + a2k
2
2
2 2
4 − 951
a y 1− 1
x y
2
1− 2
a2 + a−4 −x4ay22b 2
441
451 − 2a2 + a4 − a2b2
481 − x2y2
2 2
a b −1
2
x2 2
2
165225y
a −
− 1
−y81x
2
49 a2b2 − 1
1
25 y − 81x 53 − a2 b2
4
2
2
b2 a2
−
9
4
x 4 y 2 − x2 y 4
ALGEBRA
9 26 46 4 24k 1 2 4x24n + 4 2k4
1
a4 2n4+ 4 4 84n
b2 a22 2 24 22 2 14 2 2 12 4292 b2612 6a42 2 2 41 b222 42a24 2k 14 2 9 226n4+64 1 1
2 22
4 − xax yy − 11−− x 4 −ax1
a25by1−−1x81
yx 16a2a2−b−x22a−21b 254
y 21−61a2 − 25
x y −yx81
−yx1− x 25
a− b81x−x y −x x−yy− a b1−x y a− b−x y 56
x x− y−
by 57
a x1−−−16
4 a ba −xy− 1− 4 x 4 −a−1
b a− 1xt −−x1
y −y 55
9
4
4 169 4 4
25 9
4
4 25
16
4
4
25
16
16
4 16
2 2 22
58 x2k − 4
59 a2n+4 − 1
60 x4 − 1
61 a4x4 − 16
x4
x4
a4
a4
4 2k
n
2
3
2
8n 2
−x1 −x44 − 1 a2na+4 x−41− 16
x 4 − 1 −ay44x 4 − 16 − b4 − ty8463
− 6561− b4a2+ tb82n−− c6561
+ 2ab a264
9
+ xb22t−−
3
2yab− 1 9 x42x−4 −
y 24+x3
+
y 65
−x21−a124+x 4bx26− +−4 xc342 +
−+ x2ab
−1
yc−2 ++6561
62
16
16 16
16
674x4 − 4x3 + x2 − 1 68 x6 + x4 − x2 − 1
soluzioni
669x2 − y2 + 2y − 1
x6 + x 4 − x2 − 1
CompletaI
3
1
5
a2++ 2a
b2 ++12b
+ 2ab + 2a + 270
b
a4 + b4 + a2 b2 − 3a3 b + ab3
69 a2 + b2 + 1 + 2ab
4
4
2
Per approfondireI
x2 + 4 y 2 + 4 xy − 2 xz − 4 x − 4 yz − 8 y + z2 + 4 z + 4
EserciziI
2
2
2
2 2
71 x2 + 4 y 2 + 1+ 4 xxy2 + 4
72
2 yx 2x+2+4+1y+
4 y42xy
4+a1+2++24xbxy
++4+1
y2
+ x4+ab
4a42−y+4ba24−+a221b+ b42ab
a+21+
−+44axab
−+2−9b4+a4−
aax
2+73
b−46xa2a−2+12
+94+xx42ax
+49−++616
4
a ax
t−212
+− s6x2a+−16
412+t x16
+ 4ts24++8s16
st+t 16
+6tsx+22+4
y 216
s ++tx8+
2
2
4
4
4
2
22
2
1x 1 3 3 2x x 31 33 623
a 2 2 12 y2 y
2
2
2 2
2
2
2
2
22
2
2 2
4
4 2 22
3 24
443 4 a3
3 2 23 y3a
++74
ab
+44a−x−42+ab9
−2
+ba4ax
+ a4
− x6+a4
+−x912
x49ax
+ 4−ax
+
6a16
−−6t12
a+−xs12+x416
+ t16
4++t416
s++ts8++
st16
s +t616
+x 4tys+++48sxs75
+t +8sy6t x− 6
y4x+ yx −++4xxy
y +−y4 x− 4
y +x− b4y xy
c− 4+xy 76
+ +abc
abc
+ abc
− ay
− −−xay
bcy
+− bcy
x − x−+x −+ x a+x +−2x
b +c− b+ cay+−+bcy
4
44 4
2 4 4
24 2
2 4 44 2 2
2
a2 1 2 2 y 2 x 4 13 3 2
x14
3
2 y 3
6 3 52
4 1 2 6
5
4
2
77+ abc − x ay
cy −+ 4 xy + abc − + bayc −+bcy
+ − xbcy− x + − xa + 2ax +−3xa + −78
a −a2a++21
a + 3a − a − 2a + 1
4
4
4 2
4
2 2
44
2
4
CompletaI
27 3
1 3 2
1
1 3 31 2 19 3 2 1 27
9
79
a + a b + 9ab2a+3 +27ab23b + 9ab
– 2 + 27
a b+ 80
a y– − aay+ + a2 yy−3 ay 2 +
y
8
27
27
27
2
274
28
4
EserciziI
2 26 5 216 36
3
33
3
2 83
3 62
32 2
81 8 + 12 x + 6 x2 + x 3 8a38++12
12ax2++ 66xa2++1x82
−864
8
+a12
+x 12
96
+ 6aax222+b+6
−xa48
+1
ab82a+−364
8+b12
a33a+28
96
+ −6aa36
b+a−1248
+ 54
−ab
64
a24a+3−8+27
b96
a a 8b−−836
48
s6atab
−+ 4
+
54
s84abt434−+ 27
8sa− t36−a8s+ 54
t −a
3
27
2
1
2
1
2
7
3
2
7
1
3
8
2
1
2
7
22
3 23
2 2
3 42
36 2 6 3 42 4 4685
4
26
56 3
4
66 3
4
3n42 45 2n 2 56 n
3n86
6
23
n3n 6 n2n 4 5 n
35 64
3 46 56 331 4 55 41 16 8
56
12
4a8a3a2+3++96
612
aa+
ab1+− 6
48
−a64
ab
+ 1a23++8
−96
64
b84
aa b+
8−96
−48
36
a
ab
a
b
−
+
48
54
8
b
ab
a
−
+
8
27
8
−
b
a
36
a
8
8
+
−
s
54
36
t
a
a
−
−
4
+
27
s
54
t
a
a
+
−
27
s
8
s
a
t
t
−
−
4
8
s
t
t
−
+
x
4
s
s
−
t
t
3
+
x
−
+
s
3
t
x
−
−
1
x
t
−
x
x
b
+
x
−
3
x
−
x
−
b
+
3
x
x
+
−
1
b
b
x
x
−
−
b
b
x
x
b
−
+
x
b
b
x
−
x
+
−
x
3
3
1
80 8
80 6 2 1040 804 38 10802 10
80
140
8010 2 4 402t 14 3 5 310
1 t 5 6t 3t t
2 44t t
t 3t6tby
9 x201x−64
33 1+− a66x3yat27
32a10 +
a y+
1a−
6yxx 8+−−15
6+x8729
a 32
y a+ + a ya y+ + 332
a ay64y 2+27
+ ay8a5y4 y+
−y+15+x 27
a −y20
y +64
x t+
15
6 xx52yt64
+−
15
− 15
x 4t x8
−2t96−x520
+8x x9729
+1
3
3 3 2 3243
2 3 33 3
3
9
27
3
81
9
243
3
27
9
81
27
243
81
3
3
3
3
3
2
1t 16 6 4616t 4 4124 4 224 2 22 282 28 a 8b a b a b a b a b a ba a ba 4a 4 3 4 33 233 x23 x21 x
8
140 1 1 40 10 322n n410
7 6237
3 3 436 5 443t21t5t 4 51554t 521
85t 48634tt386 3 651
2 05
2nt n 21 4
n27
251
t 231
6166
2 x2 x−x2
3 x− 2
t 6ya4−6t+y6xa234n+ty6−x333y+nx5−x2a3n43n+yxa−1
3+
x bx−x x b+−
b−xx 4xt8t3−
x −ay2 +− 6a89
−3 + +96−a+2b−− 48
−xab−
87
15
88
1x15
a−2+
a−3
yx1 xy−+
−−bx6+xx b−−
20
x +bxx−+3xt t4bx−+6−20
15
+x−
6
+x8x−−+36
x −a+x2 +−+54
3−
xy+x63
−x+x1+
−15
b− 20
x1b−y−x56x36txbt+
−bx−15
+b−6x15
x6
y −xt 56−
+x12
x y++x6−xxxy2y+−x −xy3 x −xy8ya−x3−+y12
+ 1− −−64a+
+ 8b+
2727
24327 81
81 243 64 243
2 2 22 2
16 2
1
7927
6 6 46 4 8 4 8 8
64 648 8 89 9 9729729729 8 8 82 2 2 3 3 327 27 27 27 27
3 3
3 23
3 2 3
2
1
2
7
3
1
8
80
8
0
40
10
1
2
3
2
4
6
6
3
4
4
2
5
6
3
n
2
n
n
3
6
4
5
5
4
a
b
a
b
a
b
a
3
x
1
6a + 1 − −64a ++ 96a−b −90
48ab
+2
8b
8s91
t −32
4sa10
t + s at 8 y−+ t a6 y 2x + − 3 xa4 y+33+x − 1a2 y 4 + b x y−5 b1−x 6+x t −b15x x2−t − 20bx63tx 3+
x4 −
x 3 + 8x−2 36
− a ++ 54a − 27a
33
279
64243
8
9
729
27
81
8
27
6
4
2
2 16
1 5
4 3 10 2 4
t
2t
3t
4t
5t
6t
a y +
a y +
y 92 1− 6 x − 15 x − 20 x + 15 x − 6 x + x
81
243
CompletaI
940
95 16a4 − 8a2 − 4a
EserciziI
964a7b
9716y2
2
99 −x6 − y6
982ab − a 5
4 2
3 3
100 32a b – 20a b – 2a b –
2
3 2 4 9 5
7 6
a b + ab –
b + 2b4 – 16a3 b – b2
4
4
16
2


2

1 
1  
1
1
102
81 − a2 b + b3 − 2a3 2a − b  2a + b + b 101
x8 −
a + xb8− 81− 8a3 − b b − 1 b + 1 



2 
2  
2
4



CompletaI
44 3 2 9 2 23 2 4 3
4 2
9 2 3 2
2
b ay − a y +
a − ay
104 − a + 3ab − 2b − − a ay ++ 3ab a− 2−105
35
20
4
5
3
20
4
)
(
)(
EserciziI
106 x 4 + 3 x2 − 2 −2 + 4a3 + a2
1
2
−3 + a b + 2109
a2 −3 x 3 + xy − y 2 2
112 −2x2y
3
2
4a + a
)
1030
1 2 2
1
1
x−43++3ax22b107
−+22a2−2 +−3
4xa33 +x 4xy
a2+−3 x−23
y−+2 a b−+
22
+a42a3 +−108
a32x 3 +−xy
3 +− a2 by 2+ 2a2 −3 x 3 + xy − y 2
2
2
2
4
110Impossibile
1113a + 1
113 ab
1
9
1
9
−16 R =Q−16
a = Qa a R=a =
13 R Q
115
114 Q a = 5aQ −a13
=a5+a 14
− 13Ra =+ 14
a − R aa +=1 − Qa +x 1= x2Q− x2 x=+ x22 − 2Rx =+ 02 QR =a 0= − aQ2 a+ 4= −Ra2=+−4
= −x13=
2
2
2
2
1
91
1 9 2
9
33
117
4 xQR=x=x−=118
a−R13
= a−16
+a14R QRa =a=−−=16 aa +116
Q1R aa Q== x−a= axR+−1
a2 x= +
Q
−2x a =+R1
x=2 −0 2
Qx Q
x+ 2a= x=2R−=
a220
x ++42Q RaR===−−
13
0a2 +QQ4a x R
===
−xa−213
+R
13
xQ a Q
R= xa4=+ xa3 bQ+R ax2 b=2=a+x4ab
+ 3Q
a3+bab+4=a2aRb42+a+aab
=b−++abab424b+2b+R5
2
22
2 2
2
()
()
() ()
()
()
()
( )( )
()
()
)
( )( )
(( )) ( ) ( ) ( )
( )( )
()
(() )
( )( ) ( ) ( )
(( ))
() ()
4
3
2 2
3
4
4
5 5
5
2k
k
−13 Q ( x ) = x R ( x ) 119
= x Q ( a ) = a + a b + a b + ab + b
R ( a ) = −0ab + b ( a − b ) : ( a − b) Q ( x ) = x − 2 x + 3 R = −6
121 k = −3 h = 5
122 Q ( a ) = 2a − 6 R = 17 Q ( x ) = −3 x2 − 9 x − 2
− ab4 + b5 ( a5 − b5 ) : ( a −120
b) Q ( x ) = x2k − 2 x k + 3 R = −6 7 5
5
5
1
7 1
1 3 21 3 2
5
2
2
= 2Ra =−17
623
R=
a + aR −=29 R = 9
1
Q ( a ) = 2Qa(−a )6
Q17
= −82R =Q−124
t ) = Qt (−t ) = t R− = − R =Q−( a ) = Qa( a )−=a +aa − 2
( x ) = −Q3(xx )−=9−x3−x 28− 9 x R− 28
(82
4 4
4 2
2
4
2
4 2
4
1
7
5
1
5
Q (t ) = t −
R = − 125 Q ( a ) = a3 − a2 + a − 2 R = 9
126 a) No; b) No; c) Sì; d) No
2
4
4
2
4
1
7
5
1
5
t−
R = −
Q127
a ) =a) Ra3=−0;a2b)+ R
a −= 2−12;Rc)= R9 = ; d) R = 8x4
(
4
2
4
4
2
2
2
128 a) k = 5; b) k = 8; c) k = 4; d) ∀k; e) nessun valore di k
7
A
verificai
1 a) 6° grado non omogeneo; 5° grado non ordinato, non completo rispetto a x; 2° grado ordinato completo rispetto a y:
4 x5 y − 3 x 4 y 2 + 5 x 3
−3 x 4 y 2 + 4 x5 y + 5 x 3
b) 6° grado non omogeneo; 2° grado non ordinato, completo rispetto ad a; 5° grado non ordinato completo rispetto a b:
7a2 b3 + 2ab5 + 0, 3 − b4
2ab5 − b4 + 7a2 b3 + 0, 3
c) 9° grado non omogeneo; 6° grado non ordinato, non completo rispetto ad a; 3° grado non ordinato completo rispetto
a y: a6 y 3 + 9a4 y 2 − 0, 2a2 y − 1
a6 y 3 + 9a4 y 2 − 0, 2a2 y − 1
d) 5° grado omogeneo; 5° grado non ordinato, non completo rispetto ad a; 5° grado non ordinato non completo rispetto
2
5
4
4 5
5
2
a x: − a5 + a2 x 3 − 3 x 4 a + x5
x − 3 x 4 a + a2 x 3 − a5
3
2
9
9
2
3
)2 + ( a + b)33 (a − 41b)(2x+2 (+a2+xyb)33
3
((
)
3


 31 2
2
2  3 2
π
3 2
3 22
π
2
π +a23+− 12
− 29
2 + 3−xπ2 +b24 + π
12
π −ab
π b−22π ab
 πx a +− 2bxy  a −π ba − b 9 − π aa +− b
 2 x++28
4
2
2
4
2






3
3 33
 2 22π   2 πππ
  2 22
 1 3 33  111 2 22 3   3332 3 22 22 2333 2 22 2 22 2
2 223
22
−−babb; −c)b aaa 9
33+aaa−a−−b+bbb +++aaa+ ++bbbx2 +24
xyx xx +a)
++22xy
2xy
xyπ a − b ππ
aaa−2x
−−−bbbπ; d)
a 99
+9−−−12
πππa−a2a2
+π++12
ab
12
π2ππ−ab
ab
ab+b++2333−−−πππ
2b+bb ; e)x2
22++++ 4 +
x xπx +x++4
+44+
8++π
+ππ2xπxx+++888+++22
π2ππ
12−+−−22
3
π
;πb)
2    222
4   444 2
222
 

4
 444 
(
2 3 a − b
( ((( ) ))) ( (( )( ))
( ()
) )(
) ( (( ) ))
(
)
(
()
)
( ) )( ()
(
) ( (( ( ) )) ()(( ( ) )) ) ( (( ) ))
(
)
(
) (
)
) ( (( ) ))
5 a) a = 1 b = −2 c = 1 d = −3; b) a = 1 b = 2 c = −6 d = 1
1
7
5
5
1
Q x = −3 x2 − 9 x − 28
82b) hQ= t36;
= c) th−= ±6; d)
R =h −= ±1; e)Q ha= =± a3 − a2 + a − 2 R = 9
6 a) Rh == −1;
2
4
4
4
2
1 3
1 3
4 2
2
2
4 x22 y 4 1−2x 4 y 2 2 − 1 4
2 y
7 c, d, f; b, e; a, g
8
a + ab −4 xa4 y+2ab
+ 9
26 x−24yx4 y −+ x26
y
− t − 410
q + tq−x−t y22−t +4q4q+−tq2 − 2t−+324yq4−−272 x2−y32
15
2
2
15
83 4 2 88 42 22 82 22424 2 22 24
8
8
2
11
12 2
3
2
2
2 24 21 2 2 322
12
−+4ab
x 4 y2 −
+426
x 4xy22y+4 26 x−2xy44y11
− x−4 y 2t 2a−3 4
−
+qab
t+ tq
−−4
−4q2x2t4++ytq
4+q−26
−2t2x+24
y 4q−−
32
2−yx44−y−72
32xy −y− 72
t −x−4yq + tq
−−138a
− 2at +
b −4+q −a2
a bb +−32aaybb−−72 xa yb14 − − a4 b2 + a2 b2 − a2 b4
2
2
2
92
99
99
9
9
9
9
15
2
2
()
7
()
()
15 a2 − 4b2
16 a) Vera; b) Vera; c) Vera
 c c c c3c333c33533353
  1 1
 292929
 1 1
 29
1 29
129
5 252 52 2
3 3 3 32 3
25
32 2 2
32
A A0A
=−01=
−A01
A11A
−A
1
A1
–A
11
A−=A
−A −A−=
–=–−=– –= –=A; –AbAbA
=b=
3
3A
−bb3
5
2
+2b
b5
2
c=c−=c−c=−cc−
c+c−c3+
cc−−c+c1
;1–1
; −A11
017
A=0A
=−01
−=A1−=
−1=1
A;1=−A
11=
=−1
−1
=A1−=
−
A1=–A1–=−111
A=
–−1
=−–11
−1
=11
11
A=
Abb=
bb=3−b3=5
=b−5+
b−2;+−1bA2
A =A=
c−
325
1−bA1−A1
+−c 1
+−1c+1−c1− 1
−
bb−
+−−b+251−b2b+1
−
8A=2
888 8 8
8828 484844 4 4
 2 2 2
  2 2
 22
282
2
( () () ( ) () () ) ( )( )( () )( )( ) ( ( )( )( )( ) () )
( () )
((( )))
( () () () () )( )
( () )
( () )
( () )
( () )
111
18 a) Q
; c)Q
QQxxx ===xxx −−−xxx RRR===11;1b) Q
QQxxx ===333
xxx −−−13
13
13
xxx+++26
26
26 RRR===−−51
−51
51
111
QQaaa ===222
aa2a22−−−aaa+++333 RRRaaa ===−−3
−33
aaa+++111; Q
QQaaa ===333
aa3a33+++aa2a22−−−aaa−−− RRR===−−
333
111
444
222
111
2
3
2
3
RRR===−−− ; e) QQQ xxx ===−−−xxx+++111 RRR xxx ===999xxx; f)QQQ===bbb222 RRR===−−−ab
–––
==222aaa22−−−aaa+++333 RRR aaa ===−−−33
3aaa+++111d) QQQ aaa ===333aaa33+++aaa22−−−aaa−−−
ab
ab33
333
333
11
11
11 222
1
4
1 x = 9 x Q = b2 R = − ab3 2 – 1
2
3a4+ 1 Q Qx a= −
= x3+
a31+ aR219
Q= −xc)
−x a=−a)9 xk =R−9;
–R 20
t = −7
21R =
Q==−b)
b2 k =R −3;
ab=3k −=x + 1
−
2
3
3
11
11
2
3
((( )))
( () )
444
333
()
()
222
((( )))
((( )))
()
verso il triennio e oltrei
3
2
2
3
3
2
2
2
2
2
2
2
2
2 22
n n−2
+112+n +221nn−+−11 2=n−4−n21n − 12n2
+n1+21n−++212
nn+−−121n==−81
4nn= 4n50
2n + 2
1n 50
+−1 2–n−1−=
21n50
−=18–n1
= =82500
n 50 +50
–11+50
=12499
–50
1 =– 1
5−
1 2 2n − 1 2n + 1 − 2n −212n2−212
3
2
2
2
1 −2 1 −2 −1
2
−2
4
2n + 1 + 2n − 1 = 4
1 =Sì50 – 1 = 2500 – 16
= 2499
−1− 9801;
2 x − 10.816;
x
− 2704;
b + 1369;
a b 9216
n 4 2n + 1 − 2n − 1 = 8n 50 + 1 50
5–Sì;
49; 2401;
2
2
2 2
2
2
2
2
2
1
1
1
1−2 −2 4−1
1
1
−22 4
2n12−n+1+21n −−122n=n+−
n2n−7
1 1=50
4−n+21
21n +–=118n=−50
2n250
8
n ––1150
+ 12 –50
1− 2
=891
–x12499
= 2500
=+x2499
n+
41n1+ =28
+
n −50
–− 1+=1=2500
50
= 50
2499
1
=−–1
2500
x50
– 1−=
− −b1–−−212
a− xb −1− 2 xb−−22−+x 49a −2−b−1b−2 + a −2 b−1
; 396;
8
2
2
2
2
2
2
( ) ( )
( ( ) () ) ( () ) ) ( (
( ) ( ) ( () ( () )() (( ) )() ( ) ( )( () ) ( ( ) ( ) ) )
( ) ( )
( )( )
) )( ( ) ( ) ) ( ( ) ) ( )( ) ( )
(( ( ) ( () )( ) )
( ( )( )( ) )
SCOMPOSIZIONI DI POLINOMI
CompletaI
(
)(
) ( ( ))((
)2) (
)(
)2
4
2
4
2
5 ab a + 1 3a + 1ab a x ++11 3ya+ 1+ 1 6 x + 1 y + 1
EserciziI
4 2
4
2
2
2
4
2
7 a + 1 x2 + 2 x − 24
a + 1 x22a+−+2x1x −x1
24
−+ 2
x8
x −2
24
2− 3xy − 2
1
4−−
x xx2x +
12−5 3
x y9
− a42x−3by2 x− +a425
x+ b22x−+a25−abb+ a a−2−b+b+b24−a22ab
+b
a+23 ba−3−
2+ab
b1++a4a2 b−2b++1a34b3 +bz1aa3ab−23b2+
b
EserciziI
( )(
( ) ) ( (( ) ( ( ) ) ( () ) ( ) () ( () ) ( () ( ) ( ) ( ( ) ) (( ) () (
) ( ) ( ) ) ()((
2 22 222 22 2
3 33 33 32 22 22 2
2 2 2 22 2
x(1
x) )−(1
x )−2x()32y(−
32y4(−
x3)y4(2−
x )x4(+2xx5
−b2+ab
−a2+−abab+−+ab4−+
)10
()+25x )+(5a )−( ab)−( ab( a)− b(+a) b(+a−b2+ab
)b411
)+(4a )b(a +b(1a)+(ba1)+b(a1)+b(1a)+b1)+bz 112
)(bza −(bza2b−()a2(ab−)+2(abz)+(az )+(2z )+(2a)+((2ax )+(+axa) (+x)a +)a2 ()32x(−321x)(−3(61x)x−(−611x))(−61x)− 1)
3 3
2
2 2
2
2
2
2
+4
+)1) ( abzb( a+−12
) (ba) (ba ++z113
)) bz(2 +(aa−) (2xb)+(aa+ )z) 2 (32x+−a1))((x6 x+−a1)14
) 2 (3x − 1) (6 x − 1)
4
(
)(
) ( ( )( )(
) ( ( ) () (
) )(
)(
)
 1 1 1    1
1

1 1 2 1
2
+52xa2++22
41 a a+ −
2 2 a2a−2 +
2a2a+ +4 4 a +b 2
ab −−12
+4
b + 1  b b2 +b 1− 1  1−
b +b 1+ c 1b+2 b+ −
1c
21 5 x2 − 1 5 x2 + 1 a −52x2 −a1
 a 2
2  2
 2 4  2    2
4

) ( ) ( ) ( ) () ( ) ( ) (
) ) )
(
)(
) (12−x2b(+x −c )2(1) (+xb2 +−
( ( )( ( ) ( ) )( ) ( ) ( ) ( ( ) ( ) (
) () () ) ( ( ) ( ) ) ( () () ) ( ( ) ( ) ( ) ) ( ) (
() ( ) ) (( ) ( ( ) ) () ( ) ()) ( ( ) ( ) ( ) ( ( ) ) ( ) ( () () ) ( ) )
1  11  11   11  1  11 2 1 2 1 2 
2
2
2
2
2
2n n 2n n
−a22
a++2
a42a ++a24+
a2
+a4+a2
a−+a22
a−+2
a4
a −+24a +b 4
bb − 1bb − 1bb −+11b + 1bb+ +11
b + 1b24
1+−1b1+−cb1+1−+
cbb+1−+
ccb1−+c2bx−225
c2xx−2 22xx−2x22x+−2
x2
x ++2
x42x ++24xa+n −41
a n −a n1a+n 1
−a n1+ 1abnn++13
bn +b3
b2n −
+b3
b−b+39
b−+39
bxnn+ 9xx−n 1
23
2 22 22   22  2   24  4  4  
111  1  1 2 1 1 2 1  2 
bbb+−11 b − 1b ++11b + 1b1+
− 26
−−cc bin1++Q2
cbx2−1+cx b− −22cxx22 +
x2−
2
x2x2 +x 4
x27
−22+ 2xxa2n++−421x +a n4a+n1−1a nab−nn1
++28
13a n +b21bnn−+33bbnn+b+293n − b
32bxnnn−+ 3
x9b−n1+ 9
x29
xn2 +x x−n+
1 1xx−21+ xx+2 1
+ x +1
1bb+ +cIrriducibile
111−+ bb +1
222  2  4
2   2 4   4 

(
)(
( ) ( ) ( ( ) ) () (
CompletaI
2
2
4
4 2

2
2
1 − 2 x2b−31
5 ba +33b + 1 a − 5 b a− −51 32
b + 3Falso
a − 3 xaquadrato,
−a5+ xa − 1irriducibile
3 x a+ −
1 32x
x x Q.
−a1+ x a −3 bx +15
1 a2x
−xb− 1
30  ab − 2 x  a 3b + ab
in
3
3




2
2
2



2
2
4
4


2
4
ab − 2 x2 
a 3b + 1 33
ab − 2bx2−5
a −a 53bab+−−
−5
3 x baa+−+3
5x a −a13− 5
a −−2x
xx− 1aa+−x3ax − ba3 +x15
b3x −
x +a1
35
ab −b 2+ax323b + 1
13
x+a1− 2x
x1
+ 1 2xxa −−1b 15a − b 15a − b
115 b +ab334
3

3
 3

b −a 5− 1b + 3a − 3ax − 5
a + ax36
− 1 3 xa+−13 x2xx a− +
1 x a 3− bx +15
1 a2x
−x b− 137 a − b 15a − b
(
)
(
) ( ( ) )(
() ( ) )( ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ( a − b)(15
( ) ( ) ( ( ) ( ) ( ) (( ) ) ) ( ( ) ) () ( ) () ( )( ) ( ) ( ) ) ( () () ) ( )(( ) ) ( )( ) ) ( (
) ( ) () ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) ( ) (
)
8
)) ( (
)(
)
2
2
) ( (( ) ( ) () ) )( ( ) ( )
2
ALGEBRA
EserciziI
38 4 x − 3 y
() ( () ( )) () ( ( ) ( ) () (
)2 3 ( a − 4) ( a − 1)
(
)
(
)( ( ) ( )
) ()
3
1

49  x + 2 y  (5 − y )3
3

)(x
4
)
( ) () ( ) ) (( ) ) ( ( ) ) )(
))()
(
)
) (2x (−xy+−y1) (x(2+xy−+y156
)2 + 2(2x x−−y y+−11) (2x − y )2 + 2x − y + 1
)(
(3a −(32ab)−(12+b)6(a1+− 62ab)− 2b)
) (2x( a− −1−5( a)y(−)2a5(2)+(x2b−a+1+158
)2)b++ y1()(22xx−−(211x−) +−y1)y−2(2yx) −((213)xa−+−1y2) (b2+) x(1y−+(216)xa+−−y122) b+ ) y259
)(
CompletaI
2
1

60  ab − c + 1 3

()
2
( ) )( ( ) )(
CompletaI
55 x + y x + y + 1 (
(
3
EserciziI
57 a − 5 2a + b + 1 2
))(( ) ( ) () ) ((
( ) ( ( ) () (
(
)(
(a + b )54 (a + 1) (a − 1)2
2
+ y +1
1

3
 3 x + 2 y  50 (5 − y )
EserciziI
3
3
2
2 3
2
2
2
4
4
2
1y−y a1+x 3
+a y−y 3152
yx+2a+ −y 23
x y− xy4 +xxy+−4 +
yy 1 xx2++ yay 2+xbx2+4 +y y 4 +xa1++1
+1b2 a + b a + 1 aa−+11 a − 1
51 1− a + 3 y 1+ a − 3 y1−a +x3−
y a+a−153
4
)))(( ())(( ) () ( ( ) )
CompletaI
3
)
soluzioni
)( ) (
) ( ( ) ( ) )(( ) ( ) )( ( ) ( ) )( ) () ( ) ( ) )( ( ( ) ( ) ) ( )( () () ( ) ( ) )) ( ( ( ) ())( ( ) ) ) (((
( ) ( ) ( ) ( )( ) ( )( )( )( ) (
)( )( ) ( )( )
2
2
22
2 2
2
2 2
2 22
2
2
b() x( x +−(11x) −(3x )+( a4
x
+
1
x
−
a
+
x
7
+
a
−
9
x
+
1
y
−
1
a
y
+
+
7
1
a
y
−
−
9
2
y
+
y
2
−
1
y
4
+
x
1
+
1
y
1) ( x − 8a ) ( x + 2a )
1)x+ +
3
1
1
( ) (()y (− 1) ( y) +(1))( y −() 2) ( y)((+ 2) 47
) ((4 x +) (1) ( x ) ) ( (x + 1) ( )(−(xx2−−)8(1ya) ()+x( 2x+)+1)2 a(4)(xx2−+81)a()x( x− +1)2( ax )+48
(7) ()a −)946
(
)
2
22
1 2


2
2 2
2
2 2
1 2
12
a b− x1
3
40
4 xx +− 3y39 33a aa −− 43b a − 1 4x −x 3−x3+4y x + 3 3 3aa−xa4−−13
+ 1 x x−x2+3+ 14 x + 33a +a7−x 3−ab1− 9 x +x1−y 3
− x1 4+yx1+ 1
4

4
4


2
2
22
2
2


2 2 1 
2
2
2 2
22
2
2
2
1 2
1
a − 4 a − 1 4 x −
+ 3 y41
33a aa−−43ba − 14 xx−−3x3+y4 x + 33 a3a− 4xa −a13−b42
1x + 1xx−x+3
+1
4 x + 3aa a+ −7x3−ba1− 9x +
x1
− y3−x124+x y1++343
1 y a−x+2−71y a+−
x2+
9 1 4xyx22−++111 yx+−11a +
yx7
−+21a −y 9
+x2−
4 2
4
4 



2
2 2
2
x − 1 x + 1 x + 1 44 a + 7 a − 9 y − 1 y + 1 y − 2 y +45
2 Falso
4 x +quadrato,
1 x − 1 irriducibile
x +1
x in
− 8Q
a x + 2a
(
)
2
(x
2
)
2
(
)
2
2
 12

2
1−x3+a 2 y −c1−+ b3a− x x−+12 yc +
− y 3 + 1 ab − cx + 1
261
y − 1+x3
a− yx3 ++21y +
− 1b−+3xa+ 1 c − b − x − 1 c − b + x + 1
3

(
)(
( ) (
)( )(
) )(
)(
)
2
2
 1
2
2 1 3
+a2 y x−+12
+ y3a+ 1−x 3
+a2 y + 1c−−3ba63
− y 3x + 12 y − 1+x 3
x62
b − c + 1 ab − cx+
− x −c1− cb − bx +− 1
x +c1− b + x + 1
− y + 1

 3
(
( x + 1)
(
)(
)(
) ( )(
) (
)( )
)
EserciziI
2
2
−1+x b2a+a−x−31b−+b−c−x +
x y22+aa z−−11−+ bb −+ xx 2a − x1+ by ++ 66
xz
−x + y + z
64 3 − x a − 3b + c 2a 3− −1−x b a− −x 3b2a+3−c65
(
)(
) ( (
) ( ) ( ( ) )( ) ( ) )( )
CompletaI
67 Irriducibile in Q
(
)(
)( (
) (
)2
)
68 a − 2 a2 − a − 1
EserciziI
2
3
69 y − 2 5 y 2 + 6 y + 6 y −x 2− 25 yx2 + 2
6 y +x2670
+ 1 y x− −a2 2a5+yx24++26a y−x+126+ 1 y x−a−22a +yx44+ 2ay−3x1+2 4
+1
y71
y+ −8a2
y +a 16
y+44+ 2ay3−
+
1x 4
−y12 +y 8−
30
y2+
x 16
yx4−+123yx3+x+1−41y 2 +
x
)(
(
) ( ( )( )(
)(
) () ( )( () (
) ( ) ( ) ) ) ( ( ( ) () ( ) ( ) ( ) )(
() ( )) ( ( ) )(
) ((
) )( (
))
( )( ) ( )(
) 3 ( x − 1) 30 x ( x − 1) ( x + 1) x + y ( x + y ) (a + b) (1+ a) a − 2 (a − 2) (a + 1) (a − 3) 2a
2
2 2
2
2
2
2
2
32
M.C.D.
−++248)yya)
16
+
8 y2) y+316
3
+ (4
y−30
+1)16
x ()x30
− x1
3)((xxx−−+111))( x +30
x1+)x; y(b)
xx−M.C.D.
+(1x)y(+x y+(x)1
x)+(+ay,y+x)m.c.m.
b+()ay(+
1+b()ax)(+1+ya)a−)( a2 +a b−()a2(−12+()a()−;ac)
2+)a1)−
((aa2+−13)(()aa−−232,a))2−m.c.m.
( a12+a(12−) (a1a−−1(23)2a)(2−a12)+a21(−2) 1a +18(12)
)a −(12ya) (−a2+()y(4y2−)4(2a+)−2( y1y4)32+ 24(yyy73
(2y++4M.C.D.
)xy−231+2)(8,xm.c.m.
2
22
2
22
2
2
2
2
2
)2)M.C.D.
+1)y 30
+b2)y (1( a+
a+−1
−1)2( a(2a
2−a(3a−)−11,
a2(a−a +1+11))((2aa18
−−31
b)2 ((b22−
aa1
+)−1(1
ab−2 1
) x( ax+(+xyb−)1()1(xx+++ay1)) ( ax+−
( x−a+)2)y )(aa(−a+2+1)b( a)( a−(1−3+)2;a)d)) (2aaM.C.D.
)22)(m.c.m.
)b; +e(118
()b −(3,21a)m.c.m.
(+b1+) 1) 18b2 ( b − 1) ( b + 1)
2
2
a a + 4 a −172 y − 2 y 4 + 2 y 3 + 4 y 2 + 8 y + 16
2
2
verificai
1 a a + 1 a – 1− b ( a − b) (1− x )2 (3 y − 1) ( y + 1)(a −( zb−) (11)−( zx2)+ z(+3 2y )− 1) ( yx +−1y)3) ( x( z+ −y1+) (1z) 2 + (zx+2 2+)1) ( x(+x 1−) (yx) −( x1)+( xy2++1)4) ( x
)(
)
y( a+−1)b) (1( −
z −x )1) ( z(23+y z− 1
+)2( y) 4
+ 1)( x −( zy )−(1x) +
( zy2 + 1z) + 2)( x2 +( 1x)−( xy+) (1x) (+x y−+1)1()x52 +( 4x2) (+x1)+(2x)+( x1)−( x2−) 1) ((xx2 ++24))((xx++12) ()x(2x −−22x) + 4()x( x+22−) (xx++11)) ( x(21− 2x )+(14+) (2xx2 +− 4x x+21−) y )(1−
2
2
+−11))((xx−
1+) (4x)2( x++42) () x( x+−22) () x6− (2x) + 2() x( x++21))((xx+
−1)2( x2+−42)x( x+24−) x( x+21−) x +(11−
)72x(1) (−12+ x2)x(1++42xx2 +− 4yx) 2 − (yx) − 1−( xy−) 1( x−2 y+)1(−x22+x1+−xy2x−+yxy+ −y2y) + y(2a2) − 2(ba2) (−a24 b+)2(a24b++2a42bb2 +)
2
2
x (+11−) 2 x ()1(1−+22x x) (+
1+42
x2x −+y4)x8
−(yx)− 1−( xy )−(1x−
+y1) (−x22 x+ +
1−xy2−
x +y xy
+ y−2 )y + (ya22)9
− 2(ba)2( a−42+b2
) (aa24b++24ab22b)+ 4(bx2 +) 2a( x− +y 2+ab−) (yx ++b2)a( x+ +y 2−ab+) y − b)
1
n  n1 a − 1−n 1 b
11 a n ( a n + 7) ( a n + 1)  a −a1
(a2 − 2b) (a4 + 2a2b + 4b 210) ( x + 2a − y + b) ( x + 2a + y − b) ) a21( ya2−−1y +211)a( y−21+− y21+b1) a18( y2+−2ay2+b21)(= y2
( a 2+ 7) ( a +2 112
2
1
1
2
1 
1 
1
1 
2
2
2 2
2 2
a2 + 2a + 1 = ( a + 1)
x 3 − =  x −   x2 + x + 
a2 − 2ab + b2 − 9 =
 2 a − 1  2 a − 1− 2b13
 a ( y − y + 1) ( y + y + 1) 18 + 2a b = 2 (9 + a b ) =
8 
2 
4






  1 2 1  2 2


2
2
2
2
1
1
1
1
1
1
2 2 2
2
2
a)118
b1  aa( y− 1−−y +b1) ( y a+( yy 14
+−1y) +
y+2 +2ay2+
b21)= 2 (18
9 ++a2
ba b) ==irriducibile
2 (a92++ a22abin
+
1Q;
== (b)a +a21)+ 2a +x13 =− ( a =
+ 1)x ;−c) x3−x2 +=x +x −   xa22+− x2ab
b)3)
9 9= =( a( a− −b +
+ +bfalso
− 9a2quadrato
=−( a2ab
− b+) b−2 −
)
(
)
8
2
8
4
2
4






  2 2 2  1 


3
3
2
1  2
1
1 = ( a + 1)
x 3 − = 
x − irriducibile
x + x + in Q; d) a2 − 2ab + b2 − 9 = ( a − b) − 9 = ( a − b + 3) ( a − b − 3) ( a − 1) ( a − 3) ( a + 1) ( a2 + 1) 2 ( x − 1) ( x + 1) ( x2 + x + 1)
8 
2  
4
(
15 a) No; b) No; c) Sì; d) No
 2 2
3
3
2
1
3
3
22
22
16
M.C.D.
x
x
a
ab
+ 1x; −c)6xM.C.D.
− 2 x x+−2 1,x − 1 x − 6
2
b + 3a −a3− b a−+31 a2a+ −11; b)
+
+
−
= aa −−1,bb m.c.m.
b
b
a
b
−
= +ab−2a)−b 9
+
3
−−39 = a − 1
2 aM.C.D.
x−−31 xax++−111,xam.c.m.
++ 1
x + 12 x −
x1
− 2x +x1+ 2x2 +x −x 1
9
9
+
−
=
−

4 
( )
a2 + 1
+1
) =2
2
2n
(
(
)
)(
)(
(
)
)(
) ) ( ) ( ) ( ) ( )) ( (
)( )( )( )
(
) ) ((
))(
)) ((
) )( (
)( )( )( )(
) () ( ) ) (
)(
)(
)
2 x − 1 x + 1m.c.m.
x2 + x + 1
x − 2 x + 2 x −1 x − 6
(1+ 2)
2
verso il triennio e oltrei
2
3
2
2

 1
2
2
1 1
1
n 2 n+1
1 38 32 + 1 = 10 38 38 23
2+ 22n +=22n2+n1 9= 22ann 1
……+
++21 = 10
= 223n8 1
− +1 2
= 1=
+2
a2+n a29 + aa3n +−……
1 = 1a+n−a1 + a 2− +1 a31++……
+ a n−1 +a− 1 +1+
2  2
2
 2
2
3
10
   11 
 1
 1
1
1  1
4
= 22n 93 a n − 1 = 1+ a + a2 + a3 + …… a n−1 a − 1 1+ +   +   + ……   = 2 1−    x2 − 4 = x − 2 x + 2
x2 − =
  2 
2  2
9
 2
 2


⋅
(
(
)
⋅
((
)) ⋅
)(
(
)
()
)⋅
(
)(
⋅
( )(
)(
)
(
)(
)
)
9
2
103
11 3
11
10
  2  10
 
   11 
 1   1 1 1  1
1 2 42 
1   2  1  
2   4 2
22  4 2 
2 
2
1− +11
 4
x =+ 2x − 2
x x −+x1+ =x − x1solo
− 2 xin+
5inx R;
+ 2
+a……
=+ 2
……
=a)
+2……
−=
−1 +
1
+
x1− 4 = x −=22x21x−−+42= inxZ
x− in
2− xQxin
+ 4R
2;x=b)
− xx2− −
2x +
−Q
x1 +
−x1c)
5xx −+ 15 x −x 1+ solo
5x −in Q
5 in xR;+ 5









   222     2  
  2 
9 
3   9 3
3   9 3
3  
3 
 2   2 2 2  2

 



2 
2
2 x + 1 x − 1d) x − 5 x + 5 solo in R
− x + 
3 
3
111 333


555
111
111
333 555
111
+++
+++
5a) xxx555222−−− +++ ; b) xxx777 −−−
; c )aaa666888−−− +++
 xxx333 xxx555
222 555xxx555 xxx666
 aaa333 222aaa555 aaa666
(
)
A
(
)(
(
(
)
)(
)(
)(
)
)(
)(
)(
) (
) ((
)(
) ( ) ( ) ((
) )()(
()
)
)(
)
Frazioni algebriche
EserciziI
1 x ≠ 1∧ x ≠ 2
x ≠a 1≠∧0x∧≠b2≠ 0a ≠ 0 ∧ b ≠ 0
2
4 a ≠ 1 ∧ a ≠ ±2
5 x ≠ −3y
7 x ≠ 0 ∧ y ≠ 0
8 x ≠ y ∧ b ≠ 0
1
∧ x ≠ 2 ∧ x ≠ −1
2
6 x ≠ −1 ∧ a ≠ −b
x ≠ y ∧ b ≠ 0 3 xy ≠ ±
a ≠ 1∧ a ≠ ±2
10 −2 con a ≠ 2b
2 2 2 2
1 a − 3 3 − 3xa−x 3− 3
a2 +
x
4a
a2 + 9b2 − 6ab
3 x x3(a +4a
1a)4a
x3− 33 3(a3(+a1+) 1)
a+ 9+b9b− 6−ab
6ab
1 21
∧≠ ∧2x x∧≠≠x0−≠3−3 bab−
x ≠ 0 ∧x y≠x ≠−≠ 3
0−y3 y x ≠x −≠1
≠ a−1≠a ±≠ ± b b a ≠ 2 ∧ x ≠a −≠a3−≠2−∧2 a∧
b ≠a 0≠a 2
a ≠ −2 ∧ a con
b ≠a −≠x1−1
−∧1a∧x ≠a≠ −≠±2
11
con
b− b x ≠x 0≠ ∧0 y∧a≠y≠0≠±0 b 12
4
x −2
2a − 3b
2 x −x 2− a2 + 2 2a2−a 3−b3b
x2+ 23 a +a 2+ 2
x +x 3+ 3
+ 34
a2 4
1∧ a ≠ − b
a2 + 9b32 − 6ab3(a + 1) 3
1
a2 + 9b2x− 6ab 4a
3(a + 1)
x−3
x − 3 1 a − 3 3 − 3a
≠0
a ≠ con
b ≠−2−2
∧∧
a ≠ −2 ∧ aa ≠≠ 2
−∧
1 x ≠ −3 con
a ≠ab2≠∧2x∧≠x −≠3−3b ≠b0a −∧3 xx3 −≠3a0
a ≠ ± 13
b
± a a≠
a≠
a ≠−1−1
14
a+2
4
2a −x 3−b2
2a −23b
a+2
2
x+3
x+3 4
a2 + 2
a2 + 2
≠ 0x ≠ 1
b ≠ 0x +∧ 2x con
15
a2 + 3
a2 + 3
1 a − 3 3 − 3a
x−3
a2 + 2
x−3
a2 + 2
1
≠ 0 ∧ x ≠ 17
0
b b b≠x≠0 0∧∧xx≠≠b00
∀ a
−≠1−2 ∧ a ≠ −a1≠ 2 ∧ x ≠ a−16
3≠ 2 ∧ xba≠− 3−x33 − 3a con
2
4
x+3
x+3
4
a2 + 3
a +3
18
(
a+b
2a − b − 1
b ≠b2≠a 2a
+ 1+ 1
con
x
2a − b + 1
)
((
))
x ≠0∧a≠ −
( ( )( )
)( )(
3
5+x132 x2++33yy 5 x 4 ax axy
2 x +253xxy−3 32y +
x2n 5+xx4n +y12− 4 5 x 2 +5 x34y x2n 2+ x nax
2
2a −ab+−b1
a + b2ya −− 4b − 1
x2yn −+a4x+n b+ 1
2a − b − 1
;
;
y ≠yx−≠x5
x; ≠y −≠1−521
;1 ;
x ≠x −
≠;−5
ba + 1
0 −∧∧1ay ≠≠ −1b ;
con
∧≠a−≠b −b x ≠ 0y ∧≠ ab−5≠≠; 1−2
20
con
con
≠1±1 ; 5 ;
+1
xb ≠≠x02≠a∧0+a1
≠1
b ≠ 2a 19
5
5
5
5
5
5
5
n
n
n
y
y
1
2
a
−
b
+
1
x
2
a
−
b
+
1
−
1
x
−
x
y 1
2a − b + 1
y 32
y x 2 x + 3 y5a x2ax −
25x3x−
yx +
25
x3x+y5
5 x 5 x 25 xa 2+x1− 3 5
2 5x x + 1 5 x
2 x +1
3
ax 2 x + 3 y
axy 2 x − 3 y
x 2n + x n + 1
x
3
5x4
2 5x 2 + 3y
x ≠ −
1
;
;
;
;
a ≠ 0, x ≠ ± y
−5; 1
con x ≠ 0
22
2
a 2x − 3 y 2x + 3 y a 2x − 3 y 2x + 3 y a 2x − 3 y 2x + 3 y
5 x5 5 x5
5 x5
2 xn + 1
(
)
(
)
(
(
)
() )
(
(
)
)(
)( (
)(
) (
)(
) (
)(
)
x +(23xy )+ 3 y )
axy (2axy
x −(23xy )− 3 y )
5 x 5 x 2 25 x (52x+ 3(2y )+ 3 y ) ax (2ax
x
3 3
x
;
;
a ≠a0a≠, x≠0≠0∧,±x ≠ y± y
; ;
;
≠ −1 ;
;
;
23
con
2 2
3 xy )+(23xy )+ 3ay )(2 xa−(23xy )−(2
3xy )+(23xy )+ 3ay )(2 xa−(23xy )−(2
3xy )+(23xy )+ 3 y )
5 x5 5 x55x5 5 x5 5 x5 5 x5 a (2 xa−(23xy )−(2
2
2
2
2
6 ( x26−( xy 2−) y )
2x (2
x2x +( xy 2+) (yx −
) ( yx)− y ); ;3 y (3x2y+( xy22+) (yx2 +) ( yx)+ y ) conx ≠xx±≠y≠±y, x± y∧≠, 0x ≠∧ 0y ∧≠ 0y ≠ 0 2 (32−(3x )− x ); 5; (35−(3x )−2
24
; ;
2
2
2
2
3 ( x23+( xy 2+) (yx +
) ( yx)+( xy−) ( yx)− 3y )( x23+( xy22+) (yx2 +) ( yx)+( xy−) ( yx)− 3y )( x23+( xy22+) (yx2 +) ( yx)+( xy−) ( yx)− y )
(3 −(3x )−2 (x3)+(3x )+ (x3) −(3x )−2 (x3)+(
2
2
y2 ) ( x − y )
3 y ( x2 + y 2 ) ( x2+(3y )− x )
x xy
5 (3 − x )
x (3 +2x(3
(32++xx)2 y 26 4a x+ +23 y −23x 3 3xy−2 +2x2 y a2 +ab+22
) − x )con 4;xx≠+5±3
3(3
y − x) 3
3
x ;≠ ± y , x ≠ 0 ∧ y ≠; 0
;
0
x ≠ ± y , x ;≠ 0 ∧ y ≠25
2
2
2
2
2
(yx)+ y ) ( x − y ) 3 ( x2 + y2 ) ( x +(3y−) (xx)−(3y )+ x ) (3 − x ) (3 + x ) (3 − x(3) −(3x+) x(3) + x ) (34− x ) (3 3+ −x )x (3 − xx)32−(3y+3 x ) 2 ( a −42) x3−−yx 2 x3 −y −y 33 a2+( ab− 2
2
2
y )x (3
+
2 (+3x−) x )
x )2x )+ x2 y 5 (34−xa+x+)32y
x2
3x + x3)3xy 2 + x22y4 x + 3aya2 + 2b2 3 −
(28
45
x (+33−y x )
3 2 (x3(−3
3 xy
3−
123x xy 21+
3 x2 y 2 1
a+2
a2 + b2 −2 x 1 3 1 2
a12 + b2
x ≠ ± y , x; ≠ 0 ∧ y ≠
0
−
−29
x ≠ 0, ∧ y ≠ con
−2 xa ≠ 1,2
−
x ≠ 0, ∧ y ≠
; con x ≠ ±3;
;
con x ≠ y
27
2
2 2
2
2
2
2
3
3
3
3
3
2
3
2
y − 233
a−
( a − 2a)+ b x − ay + 12 x y − 3 3a + b
−(−3yx−) x()3 (+3x+) x ) (3 − 4x ) (3 + x3)(−3(x−
x )x )(3x(+
3−
3 −+
x )yx )(3 − x )2 ((4
3
+ 2x)) (33 −x− −xx)y (3 +2xx )− yy − 3 4 2 (aa+− b23) − x ax+−1 yx x−2 y
2
x (33xy+2x+) 2x2(3y − x )4 xa + 3
x2) x
2y5 (3 − −3
33 xy x2 (+3x+
22 yx ) a2a++b4
22x + 3 y1−2 x 313
3 xy22 1
+ x2 ya2 + b2 a + 2 1
−1
2x
3
1 2
a2 + b2
1
1
1
≠0
x− ≠a 0≠, −b∧ y ≠ −2 x
x32
≠ 0,− ∧ y ≠con
−2 xy ≠ 3,4
x ≠ 0,
; 30
; x ≠−y
con
31 2 con
23
23x−−xy
3 2
3 y23− 3
33
3 a + b2 a − a2 + 1 x −2 y
2
y
2
−
4
a
+
b
4
a
+
x
1
−
y
3
−
x
y
−
3
2
3
3
a+b
a + 1 x2
3
−
2
2
a
2
a
−
2
(
)
(
)
(
)
x
−
y
x
−
y
x
−
y
x
x
− x ) ((33+− xx)) (3 + x ) (3 − x ) (3 + x ) (3 − x ) (3 + x )
2 2
2
2
2
2 2 2
2 22+2x 21
3 −x )x )3 −2 x 3
x3(2+
3++xx)x2)y2
−
x (xy
aax+
+3
4
+b2y3 y −213x 3 313 xy
3 xy
+ 2 1 −2−x2 x1 3 3
11
4
+x2
+ x y y a +ab+a 2
212
a2a+ +
bb
11
11
x−
− 33
2x
≠ −0, ∧ y ≠ −2 x
x≠
y ≠−2−x2 x
con a ≠ −b 2 − x ≠ 0, ∧ y ≠ −
34
con a ≠ ±1
35
conx ≠
0,0, ∧ ∧y ≠
; ;
3 3 a2+2ab−
33
a) + 1x −x −yxy2 2 2 y −y33
− xx − y x232−2 y 3y − 3 2 ( aa−4+24b)
xa3−+−
y1−x x 2x x 3x−y3 −
3
−
3
a
+
2)23 3
33
a
+
b b a +a 1+ 1 x2x2
a
−
2
2
(
−
y
(
)
y
)( (+3 +x )x )(3(−3 −x )x )(3(+3 +x )x )
2 2
(9 +9 x+) (x9) (−9 x− x) ) 2 (92y (+−y1x−) (19) − x2 ) 2 ( y − 1) 1 1 2 (2y (+y1+) 1) 2 (2y (−1y t−) 2t )( y + 1) 2 ( y − t )
25 z z
t −t1− 1
t −1 (
x ≠x 0≠, −
x
t ≠t ±≠1,±±1,2±, ±
1
x ≠x ±≠ y±, xy ,≠x −≠y−−y t− tx−≠−±25
2,3± 3 t ≠ ±1, ± 2, ± 337
0con
, −∧1xy∧≠≠y90≠, −
01
, −x1≠ 0, −1∧ y ≠ 0, −1
36
con
y, x ≠
4
4
3
(
)
(
3
t +t1+ 1
8 8 y2y2
xybxyb
9 −9 x− x
x +x8y+ y
x2 x29 − x
x+y
x2
y2
2
9
9
−
+
x
x
y
+
2
1
2
y
−
t
1
y
−
y
+
2
y
+
1
2
y
2
−
+
t
1
2
y
−
t
2
1
2
y
−
t
y −y 2− 2
11
2525
z1z
25 z
1
≠x, 0
01
, −∧±
11
0−1
,3
con
con x ≠ ±y x ≠x x±≠ y≠±, 0
− t x ≠x−0≠, 0
y ,≠y 0con
z≠,≠yz0≠0,≠y,0,±1
a0≠,≠a00≠, ,xzb0≠,≠b0
±0≠,ya,0x≠ ≠0,−by≠−0t
x ≠x 0≠, t−
y∧x, ±
≠≠y 20≠
0,,,±−
1−∧1y ≠ 0, −39
1
xy ,≠−x 1
x−≠∧≠
y−y−±y ≠
ty−, 0tx,−≠
−1− y40
≠
2
− 10 8 8
y −y10
y
xybxyb
8 8 9y 2−yx28
x y+x2 y+ xy2
x+y
8
x+y
t + 1x2
y xyb
(9(+9 x+) (x9) (−9(x−92+x)2x)) (29(2−y (−xy21−)) 1)t 2− (1y − 1)
−1
2t,3≠± 3
38
±1,2±, ±
±1, ± 2, ± 3
9 −9 x− x
9 − xx 2 x 2
+1
t +1
( ( )(( ) ) ) ( ( ( ) ) ) (
)
(
( ( ) ) ( (( ( ) )) ) ( ( ) )
) )
)
(
)
(
6 6
y−y+t−y1t+ 1 2 2
y −yt − t
1− 1
4096
x
4 x6 x6 1414
1 1 2 2y +y 1+ 11 2
y −y 2− 2
27
2525
z z
44096
122y 2
y 27
−y2− 2 4096
27x27
2525
z z
4096
1
≠
≠xy−≠−xy ±t−
≠, 0y−, ≠y 0
≠con
≠≠, 0
≠0≠,,0
≠0≠,0a0,≠a0≠, b0,≠b0≠ 0
≠x,00
−1,1−
conx ≠x ≠
a ≠a 0
y x≠y ≠0
1−∧1∧
−,,1−0
± y±, yx, ≠x −
, 0zx, ≠zx0
ay0, ≠
bz0, ≠,b≠z0
≠yt±
≠
1
ytx−≠xt 0
,a≠y0
−
∧1y∧≠y 0≠, −01,−41
, x−y ,≠−x −≠y−−42
3
6
2
3
6
2
2
3
6 21212
3
y
−
10
xyb
8 8 y 2y 2 8 8 x +xy 2+
y
−
10
y yy2
xyb
+
x
y
xyb
y
−
10
x+y
xyb
64
−x21b+26164
a2bb12
b2 6464
−2b12
aa −
a1
8 8xy+x−+
y10y 8 x 8
x+x+1+xy1+ y a a−x b+
225yz−yt − 2
27
4096 x625 z 4y − 2
127
y4
−2
1 2 y +1
27
4096 x6
164
4096 x6 x 8 4
1 64
a≠≠0x0
0
≠00, a ≠ b −
≠≠00,x,y−≠1
≠ ±0y, ,zx≠≠0−, ay ≠− 0
≠43
45 a ≠ 0, b ≠con
0, ax ≠ −1,0,9
a ≠ 0, b ≠
≠≠−0y, a− ≠t 0, −b ≠ 0 x ≠ 44
, b≠, b≠−y
t , b −
x≠0
x con
,≠y ±≠yy0,≠,xz±2,10
0, y ≠ 0, z ≠ 0, acon
b −
6 xyb 2 y − 2
12 12 3
3
3
6
2− 10
2
12 12
6
24 2
12 12
3
y
8
xyb
x + yy − 10 8 x + y 3
10
a
b
64
a
b
−
a
b
64
a
b
64
−
y2
3
2
x +1
8 x+y
x +1
t8 xt ++y3t + 9 x + 1 x n −a1 − b
t 3at 2 b+ 3t + 9
xn
6
6 6
8
8 8
4096
1
6464
−2
y4−y 2− 2
27
27
x
x x
4096 x
127
4096
x x
44
1 64
0 ,46
0,≠b 0≠, a0,≠a b≠ b− −
con a ≠ ±b
con
a ≠ 0, b ≠ 0, a ≠ b47−
≠x 0≠, 0
y , ≠y 0≠, 0
z ,≠z 0≠, a0,≠a 0≠, b
≠b 0≠ 0
a ≠a 0≠, b
3 3
6
12
6 6 2 2 2 2
4
3 3
4 4
10 8 x + y 3
6412a1212b12 3
ay2 −−y10
b−210 8648xa12
+x by+ y
x +1
x +x1+ 1 a a− b− b 64
t 23+t 3+t 9+ 9
t 3 at 2 +b3t + 9
xn − 1
t 3 t t32 +
x n x−n 1− 1
6
8
8
64
4096
1 x
4
1
64
x
x
49
con x ≠ 0,±1
50 a con a ≠ ±1
a ≠ 0, b ≠ 0, a ≠48
b − a ≠ 0, b ≠ 0, a con
≠ b t ≠−0,3
4
126
4
3
a12
a2 − b2
64a12 b12 t 3 t 2 + 3t + 9 3
x +b 1
t 3 t 2 + 3t + 9
xn − 1
xn − 1
(
(
(
)
)
(
(
)
(
)
( ( ) ) ( (( ( ) ) ) ( ( ) )
(
)
(
(
)
(
) (
)
)(
) (
(
) (
)
)
((
)
)
)( )(
)
((
(
)
(
) (
)) (( ))
)
2 x + 1 a − b − a 3 − y1
2 x + 1 3 a − bb − a 3 − y
a2 b2
3− y1
3
b−a
52y −≠ 0, −2, ±3con
0, ya∧≠ay±b
, 5 x ≠ 0,±1 a ≠ 0, b ∧y−b≠≠00
, −253
, ±3, 5 x ≠con
≠≠00, b ∧
0,0−2a, ±≠3,±5b, 0 x∧≠b 0≠,
y ≠b ≠
4
2 a2 b2y yt +t 3− 3
y y t+ t3− 3
xy 2
a2 b2
x2
x2 xy 2 2 a2 b2y y + 3
2 2
1 1 a a−1−b b33 3 3− −y y
22ax2xb+2+11 aa−−bb 33−−y y x + y
−a1a 2a 2 2x x+ +
a2ab2b b b− −a a
x x++y y 3 3
bb−1
11
≠00
≠0,00
≠00
1y
con
, b∧ ∧b b≠ ≠0 0 x x≠≠00∧x∧yx≠y≠≠0≠0
y ∧ y ≠ 0−−
a ≠ ±54
b, 0 ∧ b ≠ 0 cony y≠≠00
∧3
−≠−
−,2−,2±, 3±,356
5, 5 aa≠≠±2±b,2bcon
,00
y,,−,y∧−
y−1,−2,3
0, 0∧a∧ab≠b≠0≠,0b0
b, b∧∧bb≠≠00
x x≠
, ay, y≠∧∧
yy≠y≠≠0
,x−,≠−
22
,0±, ±
3
, 5y, 5≠ 0, −21 2 aa55
,−
1∧
2
3
22
22
a + 1 x2x2a2 − a +
22
t t t−−33
44
22
ta2attb−t2b−3 3
y y y y++33 xy 3
a2ab2b
xyxy
xyxy3
x42x
3xyxyay+y1y y+ +3 3
(( ))
(( ))
10
2 x +1 a − b
1
51
con
− t ≠ 0,±3
2
t t −3
x2
(
(
(( )) (( ))
)
)
(
((
)
))
( ( ) )
(( ))
(( ))
(
( ( ))
)(
( () ) )
(
)
)
) )
xy 3
((
( ( ) )
x ≠ 0 ∧ y ≠ 060
, −1
2a
1
con
− a ≠ 0,±1
a +1
a2 − a + 3 a + 1
(
Per approfondireI
( x + xy + y )
1
2
2
2
( x + y )2
)(
) ( ) ( ) )( (
) ( ) ( )( ( )
)
( x + 1)2 ( x2 −( x2++1)xy + ya2 )(+x22(+x(2xxy+++xy1)y2+2()xy2 −) (xx++(1x1) )+21()x22a(−x+2x2−+x1)+ 1) a +a2+ 2
− 2 a) a ≠ ±2, ±5
− x ≠ 0,1;ab)≠ −±2,−±5 con
a ≠a±≠2,±±25, ±5
con x ≠ ±y
con
2
4 4
x2 ( x2 + 1) ( x + y ) 4 ( x +
( xyx+)2y()x22 + 1) x2 (xx22(+x421)+ 1)
2
soluzioni
x+y
ALGEBRA
2 2
2 2
2 2
ba− a 3 a3 ba b2
3+−1 ayb −baba−− ab 3 −3y− y b − ab − 3a 3
yx +2ya2a
12a 21a
30 ∧3 b ≠a20b2a xb +x y+ y x ≠ a02 b∧a2yb ≠ 0x, +
x +xy+ y1 12a 2a
b∧, 5
±≠
≠y57
x,3b≠≠,x5∧00≠b,b,b−
y≠∧,∧
0
b0
y0by∧≠≠ya00≠a0≠, b0x∧, b≠b∧0≠xab, ≠
b58
,y0≠∧x0b≠x≠0≠0
, 5 y ≠ 0y,a−
0, 5
, y≠y∧±≠
,0
,0
con
a5
00
≠≠
,±≠−,3b2
,∧±b3
, yby±,≠∧
ayb≠
≠1con
0,x−≠1x0≠∧0y∧−≠y0≠,−−01, −1
0
2a0, 0
y≠a∧
a∧
xb∧
≠≠b
0x59
≠≠y00≠, −022,a2,−±≠23
,±
0
, y0con
0,≠0±0∧bx,b≠≠00b∧ ay ≠≠a±−0≠b1, ,−±01b∧
0≠∧≠−0y0−≠∧ 0y, −
− −
2 2
2
3
2
4 4xy 3xy 3
4 4
3y +a322ba22b2 y y +y3+ 3 a2 b2a2xy
b xy 2
a−+a3+a3+a1+aa1++11 a2 −aa2 +a−3+aa1++a31+ 1a a+21a−2a−+
xy 3 xya3+a1+ 1 a2xya−32xy
a2 ba2 b xy 2 xy 4 4
xy 2xy
2 2
verificai
x − 4 x2 − y22
1
t − 10
1
1
x−4
t − 10
1 x2 − y 2 1
con t ≠ 3,4; b) y − 4 con
con x ≠ y ax ≠≠ −0b∧∧ym≠≠0− n
1 a)
∧ m ≠ c)
−n
2 x − 50
0 xx2 ≠− 0x −
a ≠y−≠b 0,1,2;
∧ 6y ≠ x0≠ ±5, −2, 3 2 x2 −
x x+42− x2 y 2 + y 4
t−4
m+n
x +2
t−4
2 a
4 x−y
4 x 4−−yx2 y 2m
+ +y 4n
22 22
101 1 1 11 1 1 1 11
t−−10
−−44 2 22 2 2 2 22
−t 10
−t 10
x −x4−x 4−x x4
1 1 1 11
y+
t − t10
x2 −x2yx−22xy−x2−y−2y y
≠b−≠m
−
x0x50
0xx−−xx−6
x−−x−6−xxx6
x2
≠1±,2±1,1
xyx∧
y y≠0≠00; c) 2 x22 x−2250
≠0≠0
−
2−02x50
−≠−x6±6≠x5,±
x−5
≠±,≠
±,2±5−,5
,3
, 3, 3
a≠a12≠±
a≠∧ab≠−
bb−∧
2 a)
con
x40≠x∧0
y0∧
≠y∧0
≠n∧m
−≠mn≠−;≠n−b)−nn
x0±2
−3
,±
xcon
−
≠
∧
≠
∧
0≠50
≠
2−,,−2
32
2±,
m
x ≠con
x−
−50
,5
a ≠a±≠a1, ±
a ≠a−≠ab −
m∧
2 2
22 2
4 4 24x 4
224−−
++22
−4−44
+m
n+m+
n+nn
−y+ymnm
x +x2+x 2+x x2
2
t − t4−t 4−t t4
aa1++11
2 a2+2a12+a21+
x4 −4xy4−x4y−x x−ym
x
x
xy yy+x42yy+2y4+y+4y y4
x −x xx− yx− +
(
)
(
(
)
( ( ( )( ( ) ) ) )
( ( () ( () ) ) )
2
2
3 a) 2 con a ≠ ±1; b) 2x − 50; x − 9 con x ≠ ±5, −2,3
y + 11
a2
1
x − 21
x−4
x − 41
− y2
y1+31
y + 1 3a22 2
3a2
1 −
±±1∧
b±2
∧y b≠
0 − x x+≠y ±a1≠∧y±yb≠≠∧01
0≠ 0x2 − x − x6 ≠ 0
x x∧
≠2 y±−5≠50
,−
2, 3x2 − x − 62 x2xcon
≠±0a5±≠
±
,b−≠
1−, −02, −x32con
≠ ≠b
0∧,0≠−b, 2
12
con
≠1
y ≠1
≠1
6
−a≠50
2
04
,1−,x±1,
222
,−
3 ±2
x − 6 x x≠ ≠±a5±,≠1−∧2
±,1y3
, ±≠25
ax a≠x≠±
,∧
+a y≠a2 ±≠ab±b
y∧
−
0≠,1−,0−3x22, +
−3y 2 yx a≠≠≠0±,0
−,,1−,,
2
2
4x +2
2
2
a
8 a
x
+
2
2
2
2
a
+
1
2
a
+
1
2
a
+
1
b
y +y
b
b2
2
3
+
1
1
y
a
x
−
2
1
±b ∧
b≠
x2 + y 2 y ≠ 0, −1, −2, −39
− 6 x ≠ ±5, −2, 3
±2 x ≠ ±y x ≠ ±1∧ y ≠ 1 −
a ≠a0≠, −0,−2,−3
2, −3
x ≠ ±12
, , ±3
8 ay2≠con
y≠
0,0−1,−2,−3
con
7 xa2 ≠+ ±y21, con
2
a
8
2 a +1
b2
1
x −2
−1, −2, −3
a ≠ 0, −2, 10
−3
x ≠x±≠12
, ±1,2,±3
, ±3
con
8
a
(
(
)
(
)
(
)
)
verso il triennio e oltrei
x35 + x2 −24xx +A1 x 3 x5 + Rx2 x− 4Ax x+ 1 2x−5 +
2 xx2 −R4 xx + 1
2 − 2 xR x
2 − 2x
2x
3
2Ax x
2 − Q= x +
1
−
−=
= 1+ = Q x +
=3
1+Q x +
= 1+
Bx2 x− 2B x x2 − x +x21−B2 x
B 1x
x2 − x +
x2 − x + 1
x2B+ 7x x2 + 7 x2 −
x22+ 7 B x2 + 7
x2 + 7 x2 + 7
()
()
()
()
()
() ()
() ()
()
()
()
()
()
()
Equazioni di primo grado
EserciziI
1 −2
{}
{{}
{{ } }}
{ } { {}}
{}
{
{ } {}
{{} }
163
316
1
28 25 312
28 25 12
y4= − Q
Q
tQ≠−−31,∧
2
1,t3≠ −
y 2=Q −
Q − 2 3 y = −
Q −Q −−1,21
Q−−− 22−,13,1 y =Qy−−= 4 Q −
23
3
6 27
3
6
732
28 25 12
3
5Indeterminata
6Indeterminata
7Impossibile
8 x = 5 (Dominio = Q − 2 ) y = −
Q− −
3
6
7
2
16 12 16
16
28 25 28
12 25 12
3
28 325 2812 25
3
3
16
= QQ−−Q 2−1
Q − 9
Qy −= −2 (Dominio
yQ=−− −1,=
1 Q −y =−1,1Q)
y2 =− Q
2y,−3= 2−Q −Q −2
y,Q=
t ,1
≠−10
∧yQt=−≠ −1
22, 3
−, 3,43)Q Q
−tQ
−≠−−34, ∧
30t ≠t Q
≠
−23
− ∧0Q
t ≠− −1
2 QQ−− 1−3, 3Q −
2
− Q
34−− −Q
1
Q34−∧yt−=≠1,t1−≠23(Dominio
3
3
6 37 6
7
2
3 2 6 3 7 63 7
3
2
2
3
16
28 25 12
3
− accettabile
y=
Q(Dominio
− 2, 3 = Q − 4 ) t ≠ 3 ∧ t ≠ −2 Q − 1 Q − −3, 3
Q− 0
Q −11
2 t = y4,= non
Q − −1,1→ impossibile
3
3
6
7
2
16
28 25 1612
3
y=
QQ− −Indeterminata,
2,23
t ≠− 3−∧1,t1≠ −2y= Q − 1Q − Q2−, 3 −3, 3Q − Q
Q − −1,1
13
12
yQ=− − 4con Q
4 − y0t=≠03(Dominio
∧ t ≠ −2 = Q − 1 ) Q − −3, 3
Q− 0
3
3
6 37
2
16
3
y =− 3,2,non
−
y = 14 Q
3 accettabile
Q− 4 →
t ≠impossibile
3 ∧ t ≠ −2 (Dominio
Q − 1 = Q − −3, 3 ) Q − 0
Q − −1,1
3
2
16
1
y=
Q − 2, 3 15 Qx−= 40, nont ≠accettabile
3 ∧ t ≠ −2→ impossibile
Q − 1 Q (Dominio
− −3, 3 = Q − 0 )
3
2
28 25
3
6
12
7
{}
{}
{ } { } { } { }{ } { { }} { }{ }{ }
{}
{ }
{ }
{}
{ }
{ { }}
{}
{ } { {} } { } { }
{ }
{ }
{ }
{}
{}
{ }
{}
}
{ }
{}
{}
{ }
{}
{ } } {{ }{} }} { { {} }}
{}
{ }
{}
{}
{ }
{}
CompletaI
16 a = 0: l’equazione perde di significato; a = −2: l’equazione è indeterminata; a ≠ −2 ∧ a ≠ 0: l’equazione è determinata
e ha soluzione y = 3
EserciziI
17 b = 1 ∧ ∀a: l’equazione è impossibile; b = −1 ∧ ∀a: l’equazione è indeterminata; a = 0 ∧ b ≠ −1: l’equazione è impossibile;
1
a ≠ 0 ∧ b ≠ ±1: l’equazione è determinata e ha soluzione x =
a(b − 1)
18 a = 0: l’equazione è indeterminata; a ≠ 0: l’equazione è determinata e ha soluzione x = a + b
19 a = −1 ∧ b ≠ −5: l’equazione è impossibile; a = −1 ∧ b = −5: l’equazione è indeterminata; a ≠ −1: l’equazione è determinata
b+5
a2 + a − 2
a −1
e ha soluzione x =
y = − a2 + 8a − 13 x =
x=
a +1
2a
2 a+b
20 a = 2: l’equazione è indeterminata; a ≠ 2: l’equazione è determinata e ha soluzione y = a + 1
(
)
21 a = 2 ∨ a = 3: l’equazione perde di significato; a ≠ 2 ∧ a ≠ 3: l’equazione è determinata e ha soluzione y = − a2 + 8 a − 13
22 a = 0: l’equazione è impossibile; a = ±1: l’equazione perde di significato; a ≠ 0 ∧ a ≠ ±1: l’equazione è determinata
a2 + a − 2
e ha soluzione x =
2a
23 a = 0: l’equazione perde di significato; a ≠ 0: l’equazione è determinata e ha soluzione x = 1
24 a = − 1 ∨ b = ±1: l’equazione perde di significato; a ≠ − 1 ∧ b ≠ ±1: l’equazione è determinata e ha soluzione y = ab
CompletaI
25 a = − b: l’equazione è indeterminata, ma x ≠ 0; a ≠ − b ∧ a ≠ 1: l’equazione è determinata e ha soluzione x =
a −1
(
2 a+b
)
11
A
EserciziI
26 b = 0: l’equazione è indeterminata, ma y ≠ 0; b = ±2 l’equazione perde di significato; b ≠ ±2 ∧ b ≠ 0: l’equazione
è determinata e ha soluzione y = b2 − 4
27 a = −1: l’equazione è indeterminata, ma y ≠ 1; a ≠ −1 ∧ a ≠ 0: l’equazione è determinata e ha soluzione y = 0
28 m = n: l’equazione è impossibile; m ≠ n: l’equazione è determinata e ha soluzione y = 5m − 4n
29 L’equazione è indeterminata, ma y ≠ 1 + b2
30 b = 0: l’equazione è indeterminata, ma y ≠ a ±b; b ≠ 0 ∧ a ≠ 0 ∧ a ≠ 2b: l’equazione è determinata e ha soluzione y = b
31 b = 0: l’equazione è indeterminata, ma y ≠ ± b; b ≠ 0: l’equazione è determinata e ha soluzione y = −2b
32 a = 1 ∧ b ≠ 1: l’equazione è impossibile; a = 1 ∧ b = 1: l’equazione è indeterminata, ma y ≠ 1; a ≠ 1 ∧ b ≠ 1: l’equazione
5
11
7
1
a+b
a−b
7
5
a≠ − ∧a≠ –
∧a≠ −
x=
x= a
y=
a=−
è determinata e ha soluzione y =
2
4
3
2a + 5
6
3
2
a −1
a+b
a−b
7
5
y=
a=−
a≠−
33 a = 0: l’equazione perde di significato; a ≠ 0: l’equazione è determinata eyha= soluzione x = a
a −1
6
3
2
a−b
a+b
7
5
= a
y=
y=
a≠
a=−
34 a = 0: l’equazione è indeterminata ma y ≠ 0; a ≠ 0 ∧ a ≠ −b: l’equazione è determinata
e hax soluzione
a −1
6
3
2
5 b 5 117
1 57
5 a+b a−
7
a−b a+b 7
5 1
11
7
57 a + b 11
1
x = ya = y =
∧ −a ≠ x– = ∧aa≠≠−−y =∧ a ≠ – x =∧aa=≠−− : l’equazione
= aa = − y =: l’equazione
a x≠ =− è∧determinata
a≠–
x35
ya =≠ − aè =impossibile;
∧ ae ≠
ha−soluzione x =
2
6
6
3
− 1 2 46
a −1 3
a2
2a + 5
4
3
23 3 4 2a + 523
2a + 5
Per approfondireI
1a) x = 0 ∨ x = 4; b) x = 1; c) t = 1; d) y = 3; e) x = 3 ∨ x = 4; f) t = 1 ∨ t = 2 ∨ t = 3
xx
2 x2+x 3
+ 3x +x 1+ 1= =
2424 ; b)x +x 2
+x2+x 27
+ 27
==
7272; c) x =
x=
2kg
2kg
++
2a)
22
(( ))
verificai
1a) k = 2; b) Per nessun valore di k; c) k = −3; d) Per nessun valore di k
2a) F: 3x = −4; b) F: impossibile; c) F: x = 0; d) F: indeterminata; e) F: 1 − 2x = 10; f) F: a = 1; g) F: a = 0 ∧ b ≠ −1;
h) F: 3x + 1 = −4
3Impossibile
4 Impossibile
{ } Q − {2} Q − {1} x = 163 Q − {−4, 3} x = a 2− b b ≠ 0
16
a−b
6 x = 2, non accettabile → impossibileQ (Dominio
, } = Q − {2}) Q − {1}
x=
Q − {−4, 3}
− {−11
x=
b ≠ 0 ∧ b ≠ 1∧ b ≠ 3
3
2
16
a−b
7y = 1, non accettabile →Qimpossibile
, } (Dominio
Q − {−4, 3}
− {−11
Q − {2} = Q − {1}) x =
x=
b ≠ 0 ∧ b ≠ 1∧ b ≠ 3
x=−
3
2
2
ab
2
2
a−b
3
16
a−b
3
16
x
x =Q − {−=4, Q
x = b ≠ 0 ∧ bb ≠≠10∧∧b b≠≠31∧ bx≠=3− x = − a ≠ 0 ∧ ba ≠≠ −0a∧∧bb≠≠−−a2∧
b≠
11
,Q
−, }{2} Q −Q{−2}{18
3}− {−4x, 3=})
a∧
b ≠−2−a ∧ ab ≠ − x =a
} − {Q−11
} Q −x{=1} 3 (Dominio
3 a+b
3
2b
3
2b
2
2
16
a−b
3
9a = b: l’equazione èQindeterminata;
, } Q −a {≠2b:
− {−11
Q − {1} è xdeterminata
=
Qe−ha{−soluzione
4, 3}
x=
b ≠ 0 ∧ b ≠ 1∧ b ≠ 3
x=−
} l’equazione
3
2b
2
16
a−b
3
10 b = 0:Ql’equazione
− {−11
, } èQ impossibile;
− {2} Q −b{1=} 1 ∨x b= = 3: l’equazione
Q − {−4, 3perde
b ≠ 0 ∧ b ≠ 1∧ b ≠ 3: l’equazione
x=−
a ≠0∧b≠
} dix =significato;
3
22
2b
−
ab
a−b
3
è determinata
m ≠ n∧n ≠ 0∧m ≠ 0
y=
Q − {−4, 3}
x=
x=
b ≠ 0 ∧ b e≠ ha
1∧soluzione
b≠3
x=−
a ≠ 0 ∧ b ≠ − a ∧ b ≠ −2a ∧ b ≠ − a
a+b
3
2b
2
5 y = −1, non accettabile → impossibile (Dominio = Q − −11
, )
11 a = 1: l’equazione è indeterminata, ma x ≠ 1∧ x ≠ 2; a ≠ 1: l’equazione è impossibile
16
a−b
3
2
0:∧l’equazione
,
Q − −è4,indeterminata,
11
Q− 2
Q − 112 ax ==−b: l’equazione
3
x = ma x ≠ 0;ba ≠= −b
0 ∧∧bb ≠≠ 1
b≠3
xè =impossibile;
−
a ≠ 0 ∧ b ≠ − a ∧ b ≠ −2a ∧ b ≠ − a
3
2b
3
22
3
ab
−2m − 2n
m ≠ n∧n ≠ 0∧m ≠ 0
y=
≠ 1∧ b ≠ 3
x=−
al’equazione
≠ 0 ∧ b ≠ −èa determinata
∧ b ≠ −2a ∧ eb ha
≠ −soluzione
a
x=
3
2b
m−n
a+b
ab
2
a−b
3
13bm≠ =0 n∧ =b 0:
èx indeterminata,
y≠
èx impossibile;
− −4, 3
0∧
2a≠∧0:
x=
≠ 1l’equazione
∧b≠3
=−
a ≠ma
b ≠±2;
− am∧=b n≠ ∧−m
b ≠l’equazione
− a
=
m ≠ n ∧ n ≠ 0 ∧ m ≠ 0:
a+b
3
2b
2
ab
2
−2m − 2n
l’equazione
è determinata
m ≠ n ∧ n ≠ e0 ha
x=
− a ∧ b ≠ −2a ∧ b ≠ − a
y=
∧ msoluzione
≠0
a+b
3
m−n
14 x = 1
}
{
{}
{}
{
}
}
x=
ab
a+b
y=
−2m − 2
m−n
verso il triennio e oltrei
1a) 50 °C; b) 176 °F; c) −40
133
133
x x
x x
= 15
→→
x =x 12
= 12 ; b) x +x + = 19
= 19
→→
x =x =
2a)x +x + = 15
88
44
77
Sistemi di equazioni DI primo grado
EserciziI
1 a) Impossibile; b) Determinato; c) Determinato; d) Indeterminato∀a
2 2712 27
2  1



12
2  2 12 
127
2  2 2 22  2 2 2 2
12 27
2 
 x =
x =   x = x x= = x x== x = x =x = x=  x =  x=x =  x= xx ==  x =
x=x =
xx2==  x = 2 x =x
= 9
= x10
1xx=55
1
x x= =−3
1x = 5−13
x1
=510
xx ==10
=10
1x = 155

x 13
  x 11
 1155
=−3
= 1x =510
x=
35x =10
 5
 x = 1 xx==1−3 x=x 1=−x3=13
1x x=13
3x= −13
55
x = 155
5
55
13
11
11

3
2 
        4
   5      6   
 7      5   5 
215yy196
y15= 5151y =15
y51
=2y =15y=
y15
y= =01y y= =0
y = 1015
y= 0 1
y=
==52yy==5196
2y = 2196
y5= 2 196
15
196
y =y=57y
y5=196 5
1

 y = 1 yy==10  y=y 1=y0y ==−1

y = − yy == −− y = y−=y −= − yy2== −− y = −2y =
y = − y y= = − y y==− y = y− =y = y− = y = y =y = −

165 11

165
 513 135 135
13


 5 13
11  16511
165
5  5 165
13
165  11
12
CompletaI
 x = 2
 x = 2
14 
: nonxaccettabile
(CE: x ≠ 2 ∧ y ≠ 1)
≠ 2 ∧ y ≠ 1→ impossibile

 y = 2
 y = 2
EserciziI
ALGEBRA

  12

12
2
2
12 12   27
27 
27
27
x = 2
2 x = 2
=
x x= =  x =
x2 =
xx x===  x =  x x= =
 x10
 x = 9
=1 55
x = 1 55   11
 x5= 10
55
    11  x x= =99 x = 9

 x x= =11x=x 10
55
5 x x= =10
11
11
9 
 5 5 5   5 11 
       10
  
  8
y
y
=
5
y
=
5
y
y
=
2
y
=
y
5
=
2
y
=
2
y
=
7
y
=
=
5
=
2
5
5
196
196
1
1
5
5
196
196
 y = 7
yy y===−−− y = −  y y= y= =−y =  7y y= = 7
   y =
     y y= =−− y = −
2
2
2
2




165 165 

11 11 
 
11
165
5
5
11
165
12Impossibile 13Indeterminato
soluzioni
x=
3
15
y=
13

55
5

x x==−−  x = −
x x=4=00  x =0
aa  x = 2a
x==22
x x==44  x=


x x==aa−−11x = a−x1
22 x ≠2−2 ∧ x ≠ 0 ∧ y ≠ −1∧ y ≠ −2
aa≠≠±±bb∧∧bab≠≠22
x
≠ −2 ∧ xx ≠≠ −02∧∧ yx ≠≠ −01∧∧ yy ≠≠ −−12∧)16
15 
(CE:
(CE:x ≠ y)
y ≠−2
±aab ∧ b



 


==−−
11  y =−
aa−−2y2 = 2a − 2
y==aa++11y = a+y1
y==22
y y==−−55 y = − 5

y y==11  yy=y1
y1


22
2

 x = −a − 1
a+b

 x = 4
 x = 0
 x = a − 1
 x = 2a
1
1

x =
17 
: non accettabile
→ impossibile
(CE: x ≠a0)≠ ± b ∧ b ≠ 2a
−1∧ y ≠ −2
a=−
a≠−
3






a +1
2
2
y
=
−

 y =x−=1 − 5  y = a + 1
 y = 1
 y = 2a − 2

y = a − b
2

 x = 0
 x = 4

 x = a − 1
 x = 2a
2
18 a = ±1: il sistema perde
eha soluzione 
∧ x ≠ 0 ∧ ya ≠ ±1:
−1∧ilysistema
≠ −2 è determinato
x ≠ −2significato;
a ≠ ± b ∧ b ≠ 2a


 y = 2a − 2
y = − 5
 y = 1
 y = −1
 y = a + 1

5


2
a+b
 x = −
 x = 4
 x = 0
 x = 2a
x =
 x = a − 1
2 19x a≠ =−20:∧ilxsistema
≠ 0 ∧ yè≠indeterminato;
−1∧ y ≠ −2 a ≠ 0:
a ≠ ± b ∧ b ≠ 2a
3
 il sistemaè determinatoe ha soluzione 


 y = 2a − 2
y = − 5

 y = 1
 y = −1
 y = a + 1
a+b
 x = 4
 x = 0
 x = a − 1
 x = 2a
1  y = a − b1
x =

2

perdesignificato; b = 2a: il sistema è indeterminato; a ≠ ± b ∧ b ≠ 2a il sistema
è3 determinato
−1∧ ya ≠= −±b:
2 il sistema
a = −e ha a ≠ −
x ≠ −2 ∧ x ≠ 0 ∧ y ≠ 20


2
2
y = 1
 y = −1
 y = a + 1
 y = 2a − 2
 y = a − ba + 5
 a + b 
 x = −a − 1
x
=


1
1
 x = 2

 x = 2a

 x = b − 4
x =
5
± b ∧ b ≠ 2a
a ≠soluzione
a=−
a≠−
3






a +1
2
2
y =1
y = b
 y = 3a − 10
y = −
y = a − b
 y = 2a − 2


a + 5
a +5
2









a +
b
x = − a−x 1=
a + b
a −+1b
=
−
−
1
x
=
−
a
x
a
=
x
5
a




 x = b x−
1 x = 2  x1= b − 4 
1 x = a − 1 1
 x = 2xa =

 x =1b − 4  x =1 
 x = a x−x=1=4
xx==20a
5
5
≠ b−2≠ 2a  
≠ ±accettabile);
=a−+ 1 ae ≠ha−soluzione 
∧
≠a ±≠b−∧ èb impossibile
≠ 2a   (soluzione
ab ∧
= b− ≠ 2a a ≠ − :il3sistema

3 21 a = − : ilasistema
a + 1

 èadeterminato
a 3+ 1 anon
2
2
y
=
1
y
=
−
1
y
=
a
+
1
y
=
2
a
−
2
2
2
2
2
1
=
y
=
b
y
y
=
a
+
1
y
=
2
a
−
2
y
=
b
y
=
−
3
10
3
10
a
−
a
−
=
−
y
=
−
y


a + 5  y =
  y = a − b

 y = b y






xy==−aa−y−b=
2
a + b   y = a −2 b
1
x
=
2






=a2
1
5a  x = b − 4
1
1

 x 5
x =
 x = 2a

5
22
e ha
a ≠ ±b b=∧2:b il≠ sistema
2a
a ≠ − è determinato



 perde 3significato;a b= ≠− 2: il sistema


a +soluzione
1
2
2
1
 y = 2a − 2
y = −
y = a − b
 y = 3a − 10
 y = 1
 y = b
2



5a

a +5

 x = −a − 1
a+b
 x =
1
1

 x = b − 4
x =
 x = 2
5
è 1determinato
a ≠ ± b ∧ ba≠+25a 23
a =perde
− significato;
a ≠ − a ≠ 0: il sistemaa +
3 il sistema
 e ha soluzione 
 a = 0:

2
2
2
y = −
 y = a − b
 y = 3a − 10
 y = b
 y = 1
 x =
−4
2

 x = 2
5
5a
24 
25Impossibile
26Indeterminato


 y = 3a − 10
 y = 1
5a

Per approfondireI


5
5
x =
x =
x = 1
 x2= 1
2

 

a)  y = 3 ; b) Impossibile
 y = y6= 3 ; c)  y = 6
z =
z = 5
 z = −1
 −1

 z = 5



verificai

47 
47
 x = m − n 
 x =2 2
 x =
x2
=
2
22
x =
m −≠n0 ∧ m ≠ − n
1a) Determinato; b) Determinato; c) Impossibile; d) a ≠ : determinato;
a a= ≠
a= : impossibile
3  m 3≠ 0 ∧ m ≠


3
33
3
1
y
=
y
=
1
 y = − n 

  y = 20  y = 20
2a) k = −1; b) Per nessun valore di k; c) k = −3





47
47

 x = m − n
 x = 0
 x = 2
2 x = 2
 x = 2
xx==m2 − n
 x = 0
 x = 2 x = a 3 − a
x =

2
2
y ≠ 1− x ∧ y≠
a ≠  a4Sì
=3
m ≠ 0 ∧ m ≠ − n
≠ − n
y ≠ 1− x ∧ y ≠ x
a≠
a = 3 
3
m ≠ 0 ∧ m5


3
3
3
3
 y = 1
 y = 1
 y = 20  y = − n
 y = 1
yy==−2n
 y = 1
 y = 2 y = a 1− a

y
=
20


 x = 3 z

47
47
xx == a2 3 − a
2

2
2
 x = 2
 x = 0
 x x= = m − n

 x = 2
x =2
 x = 0
 x = 2 x = m − n
1
m
0
m
n
y
x
y
a ≠ 3 a = m≠ 0∧
≠
∧
≠
−
≠
−
∧
 6 m = 0: indeterminato;
:
il
sistema
è
determinato
e
ha
soluzione
y
1
y
x
y
x
=
1
a≠
a=
≠
−
n
m
≠
−
∧
≠
3








3
3
3
n
 y = 1 x = 3  y = 2 y = − n
 y y= =20
 y = 1
 y = 1
yy == a2 1− a
 z = −5
 3
 y = 1
 x−=
a 3 − a
 x = 1
 x = m − n
 x = 0  y = 20
 x = 2




a =0∨b=0
m ≠ −n
a ≠ 0∧b≠ 0
m≠±
≠ 1− x ∧ y(CE:
≠ xy ≠ ±1)
7 
: non accettabile
→ yimpossibile
y = 1




 y = −1
 y = − n
 y = 1
 y = 2
 y = a 1− a
 z = −5


x=3
x=3
47



3
x
a
a
=
−
x
=
a
−
a
3


 x = 0
 x = 2

 x = 1

 x = m − n
 x = 0
 x x==2m − n
1∧b m
m ≠ 0 ∧ m ≠ −n 8  
y ≠ 1− x ∧ →
y ≠impossibile
x   (CE: y ≠ 1− x ∧ y ≠= 1x )
am≠≠0±∧
≠0
≠0 
: non accettabile
ay≠=01∧ b ≠ a0 = 0 ∨ b = 0
a = 0 ∨ b = 0
3
y = −n
y =1
y =1
y = y2 = a 1− a
y = −1
y y==2− n







y
a
1
a
=
−











20


 z = −5
 z = −5
x = 3
 x = 3
x = 0
x = 0
x = a 3 − a
3
=
−
x
a
a


x
=
1

2  x = 0
x
=
1
 x = 2







≠ x  y =1
a∧
= b0 ≠∨ 0
b= 0  a ≠ 0∧m
b ≠≠ 0±1∧ m ≠ 0
±1+∧2m ≠ 0
a = 010
∨ b = y0 = 1 a ≠ 0
 y ≠ 1− x ∧y ≠ x 9 y ≠ 1− x ∧ y 
3m +
 m ≠3m
2  y = 1
y = a 1− a
 y = −1
y =
 z = −5
 y = 2
 y = −1  y =
 z = −
 y = a 1− a
5

3m
3m

 x = 3

x = 3
x = 3


x
0
=



x
0
x
0
=
=

x=0
a 3 − a x x= =a 2
 x = 1
  x = 1

x = a 3 − a
3 − a

0 ∧ b ≠ 0   x = 1
y
a
b
1
≠
−
∧
≠
1
m
0
y
x
y
x
=
1
=
0
∨
=
0
a
≠
m
≠
±
∧
≠
determinato
12Indeterminato
mm
0m + 2  3m + 2 y≠x
0∨ab≠=00∧ b ≠ a0 ≠ 0∧b ≠ 0: il sistema
≠0
≠ ±1∧emha≠ soluzione
3m + 2
 y = 111 ay == 01 ∨ b = a0 =: indeterminato;

m ≠ ±è1∧
3
= 1a 1− a y y= =a 2
−1  y = −1
yy =
y =
 y = a 1− a
 z =y =
y =
 y = y = −1
1z−=a −5  z = −5
−5
3m

3m 
x = 3 
x = 3 


3xm= 0
x = 0
 x = a 3 − a




  x = 1
 x = 1
a= 0 ∨ b = 0m ≠ ±a1≠∧ 0
1−y x=∧1 y ≠ x a = 0∨ b =013 ma=≠±1
m
≠
±
1
∧
m
≠
0
è
determinato
e
ha
soluzione
0∧y∨ b=m≠1=00: indeterminato;
m ∧≠ b0≠: il0 sistema

  3m + 2
3m + 2
y =
 y = a 1− a
 z = −5
 y = y = −1
 z = −5
 y = −1
3m
3m




verso
il
triennio
e
oltrei
11 11
1A = , B −= − 2 A = −4, B = 4, C = 1
33 33
(
(
)
)
(
(
)
)
(
(
(
(
)
)
)
)
(
(
(
(
)
)
)
)
(
(
)
)
(
(
)
)
(
(
(
(
(
(
)
)
)
)
)
)
13
A
Problemi di primo grado
CompletaI
19, 10
EserciziI
2303 10
4 12; 30; 48
5 40
64 12
8 6 km; 12 km; 24 km
6 Luisa: 13; Irene: 11; Stefano:19 7
7
20
64 12
9 28
10 17
11
7
20
13 35
14 −5; 2
15 41; 43; 45
CompletaI
16 3840 a2
EserciziI
17 25°; 75°; 80°18 80 dm; 192 dm2 19 2400 dm2
12 2
20 36 dm; 51 dm
 45

a
21 68a22
48 cm; 96 cm; 26 cm; 720 cm2 23 
π + 72 dm2
 2
 2
 45

a
AH
25 16 cm; 18 cm
26 14,4 cm
π + 72
 2 24
 = 2


CompletaI
27 Perimetro: 128 cm; area: 960 cm2
EserciziI
5
8
30 43.200 dm2
28 29 Impossibile (soluzione non accettabile) 8
45
5
8
32 20; 25 33 4 m; 3 m
31
8
45
34 31 caprette; 17 galline 35 Perimetro: 80a; area: 375 a2
Per approfondireI
15; 432 53 cm
3 25 dm
verificai
1182 39, 81 3 Impossibile: la soluzione non è accettabile
4 20; 48 5
47° 30′; 32° 30′; 100°6 75 cm; 90 cm; 2700 cm2
7 48 dm; 20 dm; 136 dm; 960 dm2 8 2139 6 cm; 10 cm
10 84a; 42a
verso il triennio e oltrei
1
48 minuti; 48 km
2 t = 6; s = 12
3
1 121 8 5 8 810 1010k 2 + kb22k 2+ +b2bb22 − bk22b2− −k 2k 2
a ; aa;a a a a a a a a4
;
2 232126 1212
12 1212 2b 2b2b 2b 2b2b
Disequazioni di primo grado
{x : x
CompletaI
∧∧
x x≤ ≤−1
−1
1S = x x: x: x∈Q,
{{
EserciziI
} {{{
} } { x{ x: x: x
 
16
16 
∧∧
x x< <11  x x: x: x ∧∧
x x≥ ≥  
15
15 
 
}}
 

 
33 
x x≤ ≤− −  
 x x: x: x ∧∧
22 
 
  
 
17
17 
x x≤ ≤  
 x x: x: x ∧∧
44 
 
{ x{ x: x: x ∧∧x x≤ ≤0}0}
 
 
 
 x x: x: x ∧
 
 
11663  
3317
17
17 
1  
 
−}} xx: :xxx :∧x∧x
−≤ S=Q
: x∧∧xx∧≥≥x ≤ −
 xx::xxx : ∧x∧xx≤∧≤x−4
xx{: :x x: x∧∧x∧x≤x≤≤ 0} {{xxx:::xxx ∧∧xx≤≤00
}} {3
{xx: :xx ∧∧∧xxx<≥<1S115
}1}6=xx::xx∈Q,
44 
6  
15
152  
224 
∧2
x ≤ −S1 = xx:x:x:xx∈Q,
∧∧x∧
x≤x≤−<−111
 
  

 
 
 
 31
6  
 
17
16  
3   
117
6  
16 3
317
17 1   
11
66 3  
17
116  
−   xxx:: x:xx :∧∧x∧xxx≤≤∧
x∧ x∧≥x ≤ −1 xxx:x:x: :xxx ∧∧∧xx∧
x≤5
−1S = x x: x:x x: x∈Q,
∧∧∧x∧x∧x≤
x<
x≥<
−1
≤11 xxx::x:xxx: :x∧x∧x x∧
≥x<x 1
0
<≤−x0
1−> xx :: xxx: x: ∧∧xxx
≤≤S −0= x x: xx: x::∈Q,
> S0=  xxx:x:x: x:x∈Q,
xxx∧: ∧x∧x∧≥xx≤∧≤≤x−−−
≤  7
: xxx∧∧x≤∧x≤−x≤≤
xx x≤≤≤−01
x ∧∧∧
−<−1
≥∧6
x≤≤
x :xx: xx: :x∧
4 6   
15  
2   
154 
15 2  
15 2
2445  
15
665  
26
4 
3  17 
161   
16  17
3   
17 
17 1 
6 
1  36 


0 = xx:x:x:x :x∧x∈Q,
xx≤ ∧
x: xx ≤∧
<≤1S
∧x∧x≤∧x≤x0≥−≤1− x:xx: xx:∧xxx:∧≤
xx∧−
≤xS−>= xx: :xx∈Q,
−∧xx≤≤−1 x : xx :x∧
x<∧1
x ∧: ∧x x≥>∧ x ≤ x : x x ∧
: xx ≤∧−x ≤ 0  x x: x: x ∧ ∧
x x≤ ≤ −   x : x : ∧x x ≤∧ x0 > x : x ∧ x
∧xx8
9
4 
5  4 
4 6 
5 
2  4 
156
15
2   
6  25 


   
} { {{
}} { {
} {{
}}{{
}
14
}} {
} }
{
} {{ {
}
{
}
}}} {{
}
{
}
 
 
16  2 
3  
17
31  
∧∧x1≤<−x1 ≤ 6 x : xx : ∧4x << 1
∧ xx ≤> 0−
10 S = x : xx :∈Q,
x < 5 x : x : ∧x <≥ ∨x > x : x x∧: x ≤x −< 4 ∨ x >x6: ∧x x ∧
≠ x0≤  x : x x<: −x3 ∨
 x : xx : ∧
15
2  
3  5 
15  
4 
 
EserciziI
16  
 17
  3 16
1  2 
1   2  17 31  3  1

 
0≤ x∧
x 4≤ <−1x < 5 x12
:xx : ∧xx :<<xx1 : ∨
∧ xx∧
x−
xx ≤∧6xxx:≥: xx :<∧4∧x∨
x x0
−:∧ xx≥<
x ∨x: xx≤ >∧x−x:≤ x−<x4: xx∨: x∧
x> :6
≤3x: xxx: ∧:
:xx=6< ∧
≤x∧
∈Q,
= 6 x : xx ∈Q,
:  xx∨: <xx −>3∧
x ≠x<0
x : 11
∧1 < Sx ≤
: ∧∧
1
<: 1
4<x<1S>
5∧xxx: ≠≤
≤>
−
15  
3  5 
 4 
  2 15
3   5  4 15  2  3





16  31 
17
 
1
2 
1 3   
6 

 

:∧ x x≤ <−1
4 ∨ x >x :6x∧ x ∧≠x0< 1 x :  x x: x< −∧
3 x∨≥x > − 14
∧ x0 > −3 x :xx : ∧
 x S: x=: Φ∧−x ≤ −≤ x < 3 x: xx : ∧xx <≤− ∨ x >x6: x x∧ :x ≤
 x : x < 13∨ Sx => x: x x∈Q,
45 
15 15  
5 
3
3 2   

 
 

{
∧4 < x < 5
CompletaI
}} { {
{{
} }{ {
{} {
{ {
} {}
} {
} }
{ {}
}}
{
} {{} {
}
}
}
}
{
{
}
{
{
} }
} }
{
{} } {
}
{
}{
}} {
}
{
{}
}
ALGEBRA

 31
 
 1  2
16  
3  1 
36
17 136
17
1
31  
2 
2  
6 116    
∧−x ≤x∨
−: x x> :<6x− x∧x :∨
x4><∧xx<≤5−x1:  xx: <x 4
:xx∨< x ∧
1
>3 x:xxx:: ∧xxx<<≤
:x<xx<3>1
: ∨x x≤∧>xx6≤<0
xxx≤−3
x: x≤x∧>x∧−
15
S−x=≤x:x : x x<
∈Q,
>Sx =≠0x x:x:x ∈Q,
:∧
x x<x≥4<∨−3
xx∨
x:∧x≠x ≤
0∧−x16
3xx: :xx∧xx:∧: >
>∨x6x<∧1
> :6
xx∧
>x −
x∨
:∧− x −≥
<−
>x−
3x∧
∧
≤xx :−: x−x: <x −≤
15   15   2  3 
25
4 755
3
5 
5  
5 315    
 4 

 15
 6  5
}{ {
{
17
1    16  7

   31  16 6  1 
  6  
2 
13 
2
31   
1

  13
  3    6
x∧
x−43∨∨
∧xxx>≤>6−−∧
1 x ≠ 0x:xx: x :−∧
x x<≤<1Sx−=<
x x≤
x: ∨x∧
x−1
17
Sxx<=>≤−6x:∨xxx∈Q,
:∧ xx≥< −x : ∨−x x>:≤
6xx <∧xx3: ≤∧−x x>:−3x <x−:x18
:>∧∧x−xx≤: >
−
∨∧ x≤>x x−
: : x2 ≤∧xxx≤≥x: 3: x∨ <x ∧0
≥x∨>xx :>xx1 : ∧∧
: >−2
3
∨x:−xx<>11
3∨
 xx: >x x−∈Q,
∧3xx≤:x0xx:: x< x0 <
54
75 
5
75   15  2
15   
3

   15  15
 5  

  6
 5 3 
  2    5
 
} {}
}
{
{ { {
}
}{} { }
}}
{
{}
} {
{
}
}
}
}
} { { {
} { }}{
}}
{
{ { {{
} {}
}{
} {
}}
{ { {
{}} { }
}} {
{
}
{
{} { {
} { }}{
}{
}{
}} {
{ }} {
} { }
}
{
}{
{
}
soluzioni
16 2
17 
 6
 
13   3   2
7   
31  
6 1

 13
 1  7 161   6  
  
xx : ∧ x x≤< :−1− ∨ x≤x>:x6x< 3∧
x:<x∧1x:S >=x−<
1∧x∨x≥x≤ >x01: x x x: x∧: ∧
x: x2< >1
x :3∧xx >∧
x :x20
x19
S
x∨∧
: xx x>≥<6 − x ∨: x∧x>x: −
∨x>x1∧
> x=−x≤ :x : 2x ≤∈Q,
x∧
x: xx≤x≤:3<x−∨0
≤∧3xx ≤≤3x−∨
: xx−≥1∧≤xx≥x≤: xx :  ∧∧xx>
:>
x 0x:
3x −: x ∈Q,
>x −3∧x ≤
xx−:: xx << 0−∨
15
2   5  
75   2   5
2   
5  
5 3
 5
 

 754 
 6  2 15
  
 5

}≤−−1x

 116   6 
7   3   
1   1 
13  
2   13  16  
7 17
 
  1 

 2
∧x∨≤xxx:−=: ∧
x− ∧x x∨: ≤x∧−>
x<<<0−1S∨ =x∨>xx1>: x−
1∧∧xx>≤x 0
x0
x1x≤:x∧xx: ≤x<> 110
x5x∨>
∧xx0=≥x : x :x∧x:x>>x10
x−3
xx21
S−x≥=
x :<x∨3 x∧∧=x 5≤≠∨−
:∧ x2≥≤x :x ≤x 3<∨0xx∨:≥x > ∧
x >≤ 3x x≤:22
1≤≤ xx :≤xx ∈Q,
: ∧∧xxx≤: :>
: :−
1x x≤x:−: ∧ 2
3x ∨: x∧
x1>− −3 x :xx : ∧
 x∈Q,
2   2   
2   6 
75  
5   75  15  
2 4  2 
 215
 
   5 

 5
16  17
3  
 1  1   16   6 


117
1 
13 
7 

 
 

x∧xx: ≠∨
x<−x13∧=x5≤∨xx−x:: =
x x: >∧2x−≤≤ x−1
xxx <:>11
xx x≤: >
xx <∧0x∨≥x1: ≤x xx ∧
x∧ :xxx≥≠ 3−∧3x ≤x−:x
xxx≤>>−03 xx :: xx−:24
x−1
23
S∧=x>x x3
≤<x23>∨
: : x2∈Q,
x≤∧: xx ≤
−≥13≤∨xx≤≥ x: xx: : ∧∧∧
>1x 10
≤∧
xS∨x≤=x≤ =x5: x∨
x ∈Q,
:x ∧=∧
0∧x x:≤x x0: :<x3>∧
∨
10
≤ 3x∨: x x:≥x< 0∧
15  22
2  
2 4 
2 
75 
2 
 6 2   15   5 



 

 1 


 16   6 
1 
1  16  
117
7 



 1  3  
 
x>≤ 33∧∨xxx≤≥:−1−1 ≤xx: ≤
: x∧ x >∧
x3 <
1∧
x2 ∨x∧xx: :≥x<3x1∧<x0≤
∧≤ x∨2≥x∨ :=xx0≥3∧ x>xx: :x x ∧>x2≤∨−x
x5∧∨x∧
≤3x : xx∈Q,
25
26
Sx≠x==−
: 1S∧
x=
x: >−x01:≤xx∈Q,
x ≤∧: x x≥> 10
x : ∨ ∧xx=x>5: 0x∨ x∧=xx : ≤x−>x10
:∧ xxx≤:<<x−03
∨∧x1x≤≠x0
x −:≤3
x∨x:−1: x≤x>x 2
: ∨xxx< =:3
2 
2  15  
24 
2 
 6 

 15   5 




 2  2  
16  
3  
16  


17 
1 
 1 
∧
1 ∧∨ 1
∧xxx: <≥x13< 0∨
∧
: x ∧x ≤ 0x : x  x∧: xx ≤ −
x => 50 ∨ xx=: x> 10
x : ∨ xx <=27
35 ∧∨Sxx =≠= −x3: x x∈Q,
x:∧: xxx≤<<−30
x ≤≠x x−: 3x≤ 2 ∨
xx1
: : ≤xx x>28
≤2x2∨≥∨Sx x==≥0
x :∧: xx x≤∧>−x1
2≤∨−x =x:0x  x∧:xx < 1∧ x ≤ x : x ∧ x ≥
x3 : x x∈Q,
15  
2  
15  
2 


 4 
 2 
} {} {
{
}
}
}{ {
{
}}
}
} {{
{} }
}
{ } {{
{
{
}{
{{
{
} {
CompletaI
∧∧x −≤1−<1 x < 2x : x
29 SS== xx: :xx∈Q,
EserciziI
{{
}
} {}
}
∧x < 1

x : x

∧x ≥
16 

15 

x : x

∧x ≤ −
3 

2 

x : x

∧x ≤
} }{ {
}} {
{
17 

4 
}
{x : x
∧x ≤ 0

x : x

}
∧
{{
} }{ {
} }{ { {
}} {{
}} {}{{ { } { } } {}{ }{ { } }{ }}{ {
}{} {} { {
−1<≤} xx {<<x00:}x∨ x∧{≥x 5:<}x1}34
∧:>xx∧2≥x} −=1{5}0x}≤:{xx{<x: x0
∧<x∧3≥
<xS∧1
5>}=}x2Φ
<∧ x <S10=
x∧ x∧ ≥− 5
1
:∧xx∨35
{ x∧: xx <∧1}x ≥{−x1:}x { ∧x :−x133
}} {{x x{: x:x x:∈Q,
{−x{5x: x≤: xx∧<∧x0=−∨10<x} ≥xS{5<=x}0:{}xx{ x: x∧{: x∈Q,
}≠{{x0x{::∧xxxx: x≠∧∧2−x}∧1=x<0<x} 3<{∧0x}x: ≠x {0x∧∧:xx>≠22−}}5 {≤xx
∧
x
:
x
∧
x
<
1
x
:
x
∧
−
1
<
x
<
0
x
:
x
x
∧
:
x
x
≥
−
1
5
≤
x
x
<
:
x
0
∨
∧
x
≥
<
1
5
x: <
x 0∧∨ −x1≥<5x} < {0x} : x {36
x∧: xx =S0−=}5{≤x{ x:xx: <x∈Q,
0 x∨
∧≥xx −>≥12
5
x
x
:
:
x
x
∧
∧
x
x
=
<
0
3
∧
x
≠
:
0
x
∧
x
∧
≠
x
>
2
2
x
:
x
∧
x
<
3
∧
≠
∧
≠
2
S
}} {37
} }{ {
}} {{xx :: xx ∧∧−x 1=<0x} <{0x}: x { x∧:xx> 2−} 5 {≤xx
{
} }= {}{ { ∈Q, } {
:xx: x∧ x∧<x 1< 1 x :xxx: x: ∧x −∧∧1−x<1≥<
xS−<x=10
:xxx∧: ≥x≥ −<−115− ≤
xxx<x:: x0
∧∧∧x∨x −≥
<x<S11
5≥
x∧≥
−
=
<<0
x00x:≤x:xx:∧<xxx∧0:<:x∧
−−255x ≤:≤xxx
x∧ x∧≥x −≥1−1 x31
< x0x ::xxx: x∈Q,
x∧∧
5 ≤
<
x∨
0
1=<5xx<xx: :0x:xxx∈Q,
:∧x∧
x−∧1
:−
1xx1
=xx0−<<5
∨xx1>xx2>≥
30 S = x :xx: ∈Q,
32
Per approfondireI
2a −26
2a −26
a−6
1 1
1
a−6
a < a2a <:<x22:<: x <
a = a0 =∨ 0b ∨
= b0 = 0a ≠ a0 ≠∧ 0b ∧
≠ b0 ≠ a0 > a− b> ∧− ba ∧ aa < a− b< ∧− ba ∧ aa = a = a < a <∨
1a = 2: ∀x ∈Q;
;aa>>a22:>: x2 >: x >
a−2
a−2
3 3
3
a−2
a−2
2a = 4: la disequazione non è mai verificata; a < 4: x < a − 1; a > 4: x > a − 1
a−3
a−3
2a − 6
1
−6
1
1
x > 1+
< a < 1: x <
a < ∨ a > 1: x >
a>2: x >
a ≠ 0 ∧ b ≠ 0 perde
a > −dib significato;
∧a
a < −b ∧ a a =
3 a = 0 ∨ b = 0 : la disequazione
34a − 1
a−2
3a − 1 a 3− 3
22a − 6
3
4 a
a 3− 3
1
1
1
x > 1+
x < 1+
< a < 1: x <
>
a = 0 ∨ b =
a < ∨ a > 1: x >
0
a ≠ 0 ∧ b ≠ 0 : - aa >= −−b:
b ∧laadisequazione
a < − b ∧ anonaè =mai verificata
3aa−−31
3aa −−31
31
31
31
2aa−−26
a42
a42
x > 1+
x < 1+
< a < 1: x <
a < ∨ a > 1: x >
0
a ≠ 0 ∧ b ≠ 0 - a > − b ∧ a e baconcordi
< − b ∧ ax > aab=
>
a = 0 ∨ b =
3aa−−31
3aa−−31
3
3
3
1
1
1
2aa−−26
a42
a42
x > 1+
x < 1+
< a < 1: x <
a < ∨ a > 1: x >
0
a ≠ 0 ∧ b ≠ 0 - a > − b ∧ a e badiscordi
< − b ∧xa < ab
a=
>
a = 0 ∨ b =
2
3a −41
3a − 3
3a − 3
a−2
4
36
−
1 3a − 1
1
1 3
a
a92 a
≠ 0 a > − b ∧ a - a < − b ∧ a e ba concordi
x > 1+
x < 1+
x<
=0∨b= 0
< a < 1: x <
ax<< ab∨ a > 1 : x >
a ≠ 0∧b
=
3aa−−31
3aa−−31
3
3
3
a −−14
36
9a
1
1
1
a42
a42
≠ 0 a > − b ∧ a - a < − b ∧ a e ba discordi
x > 1+
x < 1+
x<
=0∨b= 0
< a < 1: x <
ax >< ab ∨ a > 1 : x >
a ≠ 0∧b
=
2
2
3
3
3
a − 14
4
2a − 6
2a − 6
2a − 6
a −a 3
a−3
131a − 1
1
13a − 1
1
a − 3 a4 1
0 a >aa2=≠:1:0
∧ bdisequazione
≠ 0 a >a−=b perde
0∧∨a b =dia0significato;
< − ba ∧
≠ a0 ∧ ab ≠
= 0 : la
a>
− b ∧∨aa > 1
a :non
<x−>bè∧mai
a verificata;
a = < a a< <
> a < 2 : x a< = 0 ∨ b =4
>
<
adisequazione
1 : x ∨< a > 1 : x > x > 1+ ;
x la
<xa<<11+: x <
a1− 2
a−2 a−3
3a−−91a 33
a−2
3 − 9a
3a − 1 a2 3
3 3a − 1
a2
36
4
43
36
1
a−3
< a < 1 : x <
x>
a < ∨ a > 1: x >
x > 1+
x < 1+
x<
3a − 1
a − 14
3a − 1
3
3
a − 14
a2
a2
a − 3a − 3 1 1
1 1
1 1
9a− 9a
a − 3a − 3
4 4
4 4
36 −36
9a− 9a 36 −36
perde
< adi <significato;
a <∨ aa>∨=1a0:: >xl’equazione
1a: <x 1<: x <a > 0: x > 1x+> 1+; a <x 0:
1>: x >
−ab<∧−ab ∧ a = a = a < 5
x> x>
< 1x+< 1+ x < x <
2
2
2
2
3a −31a − 13 3
3 3
3 3
a − 14
a − 14
3a −31a − 1
a − 14
a − 14
a a
a a
4
4 36 − 936
a−3 a−3
−13
1
4
4
a − 9a 36 − 936
a − 9a
1+ x < x < ; a x> >14: x >
> 1∀x
+ x∈Q;
x >a1=+ x14:
< a < 1<: ax < 1 : x < 6
< 1a+ x< <14:
a −31
3
3a − 1 3a − 1
a2
a2
a2
a2 a − 14 a − 14 a − 14 a − 14
verificai
16 
4  
1



: xV; ∧
: x x <a2=∨−2
∧ x se
≠ 5x > 0;g)
x ≥si ha x < 0;xf): F:
x solo
x : F:x non
x <è −mai
− < x < 0 ∨ x > 2
3 ∨verificata
1a) V; b) F: ∀x ∈Q − −2 ; c) V;x d)
e)xF:≤a =2 mai
 x verificata;
5 
3  
2





16     
16 
4   
44   
11
 
 16
 4   16 
1
55
−2
−2
∧∧x x≤≤   3
: x ∧x −
≤x2xS: x:=x ∧x∧xx: xx≠: ≠x∈Q,
∧ x x≤<x 2
x:x:∨xx x≥x<:<−x3
−3∨∨x−−<x 2:<x∨
<xx<∧≥<x00≠∨∨5xx>>22
−3−3∨3<−<xxx≤:<≤x−x2
−2<
2
S =  x x: x: x∈Q,
 x x: x: x x−x<2<22∨∨x x≥≥
xx: :xx x x∧:xx: <x≠−5
5     
33   
3   
22
 
 55 
 3   5 
2
{ }
{{ }}
{
{ }
{{ { }
}
{
}}
} {
}
4   1
44   
16 
4  
16  11
3316 
   
 16
 
88
   
x ≤:−x<2
−3∨∨−− <
x <:xxx<<0∧0
−
∨
x2
∨≠xSx5
>=>22
: x: x≤<−−333<∨<x−x≤
x ∨<xxx0
∨21
x>33
−2∨
=<=1
∨x∨∨−>x−2≥≤≤x x≤x≤2:2∨xx∨x:x>−
x xx =:<x1<x∨x
5
   x x: x: x x x<<−222∨∨x x≥≥x4: xS=∧ xx ≤x: x: x∈Q,
∧∧x x≠x≠5:5x xx x<: x:2x∨xxx<≥<−3
x: xx:∈Q,
xxx∧
∧<xx≠≤x5−x: 2x: ∨
3   2
33   
3  
5  22
22 5 
   
 55 
 
55
   
16 
1
3
8
4  
2  
 
  
 1

x < −3 ∨ − < x < 06
∨ x−2>S2=  x: xx : ∈Q,
x∧ x −≤3 <x ≤−x2: ∨x x =x1<∨2−∨ x≤≥x ≤ 2 ∨ x x> :3x ∧xx≠: 5x
x
<x :x x≤ 2x∨<x−≥3 3∨− <x :x x< 0 ∧∨xx≤> 2 xx :: xx ∧−x3><2x ≤ −
2
5 
3  
2
2
5 
3  
 
  
 2

{ }
{{
}}
{
{ }
{ }}
{
{
}
}
{
}
7 a = 3: la disequazione non è mai verificata; a > 3: x < a + 3; a < 3: x > a + 3
16  2   
3
1 1
7 16    1 
7
8 4      8
 
 3
 42  
 
5∧
∨
x>x≤2
−
:∈Q,
x∧x x≤≤2 <∨ xx ≤> 23
−x23S:x= x∧x:xxx: :x≠x∈Q,
x∧
− x< >2
x<2
<xx<≥∨
< 2 x∨ <
:xxx≥ ≤ ∧2x∨≤x ≥
∧
0 xx∨:>xx3>2−x : x<xx∧: <x ≠ 5
−∨3x <>x3x ≤: x−2
2∨∨−xx2>S=3=1∨ x−: x ≤
 x∨ : xx ≥
2 ∨xx =: x1∨ − 3 <≤ xx ≤≤ −28
 x3:xx9
≤x :x x x: xx<x: −:x3x−∨ x∧
5  3   
5 3      5
2
2 2
3  5    2 
3
 
 2
 33  
 
{ }
{{ } {
} }
{
}
{
16 
11
77
14   7
282 
2 

   
     
   
x−3
∧ x−≤ < x < x : ∨x x >x3< 2 ∨ x ≥12 S = Φx : x
≤2≤2∨∨
xx ≤x≥2
≥3∨
x≤≤ S =< x x≤x:2x: ∨x∈Q,
x∧∧
≥x x3
x x: x: x: x∧ x−−≤ <<x x<<x :∨x∨x x>∧>3
3
>>
>2S2 =  x x: x: x∈Q,
 x > x3x: x: x  x∧∧x:10
22 11
5 
23   3
223 
33
353 

   
     
   
{{
}}
{
{ }}
{
}
}
∧x ≠ 5
verso il triennio e oltrei
1 a) Se a < 0 allora a2 = a ⋅ a > 0; b) se a2b > 0 allora b > 0, visto che a2 > 0
1 1
allora a > b
2 Falso: se <
a b
3 a) F; b) F; c) F; d) V; e) F; f) F
15

x : x


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