# 8o ano matematica gabarito da bateria de exercicios

Propaganda
```GABARITO DE MATEM&Aacute;TICA 8&ordm; ano
Produtos Not&aacute;veis
Produtos Not&aacute;veis : exerc&iacute;cios com gabarito e teoria
H&aacute; certos produtos que ocorrem freq&uuml;entemente no calculo alg&eacute;brico e que s&atilde;o chamados produtos not&aacute;veis.
Vamos apresentar aqueles cujo emprego &eacute; mais frequente.
Observe: (a + b)&sup2; = ( a + b) . (a + b)
_______________= a&sup2; + ab+ ab + b&sup2;
_______________= a&sup2; + 2ab + b&sup2;
Conclus&atilde;o:
(primeiro termo)&sup2; + 2.(primeiro termo) . (segundo termo) + (segundo termo)&sup2;
Exemplos :
1) (5 + x)&sup2; = 5&sup2; + 2.5.x + x&sup2; = 25 + 10x + x&sup2;
2) (2x + 3y)&sup2; = (2x)&sup2; + 2.(2x).(3y) + (3y)&sup2; = 4x&sup2; + 12xy + 9y&sup2;
Exerc&iacute;cios
1) Calcule
a) (3 + x)&sup2; = ( R: 9 + 6x +x&sup2;)
b) (x + 5)&sup2; = ( R: x&sup2; + 10x + 25)
c) ( x + y)&sup2; = ( R: x&sup2; + 2xy +y&sup2;)
d) (x + 2)&sup2; = ( R: x&sup2; + 4x + 4)
e) ( 3x + 2)&sup2; = ( R: 9x&sup2; + 12x +4)
f) (2x + 1)&sup2; = (R: 4x&sup2; + 4x + 1)
g) ( 5+ 3x)&sup2; = (R: 25 + 30x + 9x&sup2;)
h) (2x + y)&sup2; = (R: 4x&sup2; + 4xy + y&sup2;)
i) (r + 4s)&sup2; = (R: r&sup2; + 8rs + 16s&sup2;)
j) ( 10x + y)&sup2; = (R: 100x&sup2; + 20xy + y&sup2;)
l) (3y + 3x)&sup2; = (R: 9y&sup2; + 18xy + 9x&sup2;)
m) (-5 + n)&sup2; = (R: 25 -10n + n&sup2;)
n) (-3x + 5)&sup2; = (R: 9x&sup2; - 30x + 25)
o) (a + ab)&sup2; = (R: a&sup2; + 2a&sup2;b + a&sup2;b&sup2;)
p) (2x + xy)&sup2; = (R: 4x&sup2; + 4x&sup2;y + x&sup2;y&sup2;)
q) (a&sup2; + 1)&sup2; = (R: (a&sup2;)&sup2; + 2a&sup2; + 1)
r) (y&sup3; + 3)&sup2; = [R: (y&sup3;)&sup2; + 6y&sup3; + 9]
s) (a&sup2; + b&sup2;)&sup2; = [R: (a&sup2;)&sup2; + 2a&sup2;b&sup2; + (b&sup2;)&sup2;]
t) ( x + 2y&sup3;)&sup2; = [R: x&sup2; + 4xy&sup3; + 4(y&sup3;)&sup2;]
u) ( x + &frac12;)&sup2; = (R: x&sup2; +x + 1/4)
v) ( 2x + &frac12;)&sup2; = (R: 4x&sup2; + 2x + 1/4)
x) ( x/2 +y/2)&sup2; = [R: x&sup2;/4 + 2xy/4 + y&sup2;/4]
Observe: (a - b)&sup2; = ( a - b) . (a - b)
______________= a&sup2; - ab- ab + b&sup2;
______________= a&sup2; - 2ab + b&sup2;
Conclus&atilde;o:
(primeiro termo)&sup2; - 2.(primeiro termo) . (segundo termo) + (segundo termo)&sup2;
1) ( 3 – X)&sup2; = 3&sup2; + 2.3.X + X&sup2; = 9– 6x + x&sup2;
2) (2x -3y)&sup2; = (2x)&sup2; -2.(2x).(3y) + (3y)&sup2; = 4x&sup2; - 12xy+ 9y&sup2;
Exerc&iacute;cios
2) Calcule
a) ( 5 – x)&sup2; = (R: 25 – 10x + x&sup2;)
b) (y – 3)&sup2; = (R: y&sup2; - 6y + 9)
c) (x – y)&sup2; = (R: x&sup2; - 2xy + y&sup2;)
d) ( x – 7)&sup2; = (R: x&sup2; - 14x + 49)
e) (2x – 5) &sup2; = (R: 4x&sup2; - 20 x + 25)
f) (6y – 4)&sup2; = (R: 36y&sup2; - 48y + 16)
g) (3x – 2y)&sup2; = (R: 9x&sup2; - 12xy + 4y&sup2;)
h) (2x – b)&sup2; = (R: 4x&sup2; - 4xb + b&sup2;)
i) (5x&sup2; - 1)&sup2; = [R: 25(x&sup2;)&sup2; - 10x&sup2; + 1)
j) (x&sup2; - 1)&sup2; = (R: x⁴ - 2x&sup2; + 1)
l) (9x&sup2; - 1)&sup2; = (R: 81x⁴- 18x&sup2; + 1)
m) (x&sup3; - 2)&sup2; = (R: x⁶ - 4x&sup3; + 4)
n) (x – 5y&sup3;)&sup2; = (R :x&sup2; - 10xy&sup3; +25x⁶ )
o) (1 - mx)&sup2; = (R: 1 -2mx +m&sup2;x&sup2;)
p) (3x + 5)&sup2; = ( R :9x&sup2; + 30 x + 25)
PRODUTO DA SOMA PELA DIFEREN&Ccedil;A DE DOIS TERMOS
conclus&atilde;o:
(primeiro termo)&sup2; - (segundo termo)&sup2;
Exemplos :
1) ( x + 5 ) . (x – 5) = x&sup2; - 5&sup2; = x&sup2; - 25
2) (3x + 7y) . (3x – 7y) = (3x)&sup2; - (7y)&sup2; = 9x&sup2; - 49y&sup2;
EXERC&Iacute;CIOS
3) Calcule o produto da soma pela diferen&ccedil;a de dois termos:
a) (x + y) . ( x - y) = (R : x&sup2; - y&sup2;)
b) (y – 7 ) . (y + 7) = ( R : y&sup2; - 49)
c) (x + 3) . (x – 3) = ( R: x&sup2; - 9)
d) (2x + 5 ) . (2x – 5) = ( R: 4x&sup2; - 25)
e) (3x – 2 ) . ( 3x + 2) = ( R: 9x&sup2; - 4 )
f) (5x + 4 ) . (5x – 4) = ( R: 25x&sup2; - 16)
g) (3x + y ) (3x – y) = (R: 9x&sup2; - y&sup2; )
h) ( 1 – 5x) . (1 + 5x) = ( R: 1 - 25x&sup2; )
i) (2x + 3y) . (2x – 3y) = ( R: 4x&sup2; - 9y&sup2; )
j) (7 – 6x) . ( 7 + 6x) = (R: 49 - 36x&sup2;)
l) (1 + 7x&sup2;) . ( 1 – 7x&sup2;) = (R: 1 - 49x⁴)
m) (3x&sup2; - 4 ) ( 3x&sup2; + 4) = ( R: 9x&sup2; - 16)
n) (3x&sup2; - y&sup2;) . ( 3x&sup2; + y&sup2;) = (R: 9x⁴ - y⁴)
o) (x + 1/2 ) . ( x – 1/2 ) = ( R : x&sup2; - 1/4)
p)(x – 2/3) . ( x + 2/3) = ( R: x&sup2; - 4/6)
q)( x/4 + 2/3) . ( x/4 – 2/3) = (R: x&sup2;/16 - 4/9)
4) Desenvolva os seguintes produtos not&aacute;veis abaixo:
a) (2a+3)&sup2; = (R: 4a&sup2; + 12a + 9)
b) (2 + 9x)&sup2; = ( R: 4 + 36x + 81x&sup2; )
c) (6x - y)&sup2; = (R: 36 x&sup2; - 12xy + y&sup2;)
d) (a - 2b)&sup2; = (R: a&sup2; - 4ab+ 4b&sup2;)
e) (7a +1) (7a - 1) = (R: 49 a&sup2; -1)
f) (10a - bc) (10a + bc) = (R:100a&sup2; - b&sup2;c&sup2;)
g) (x&sup2; + 2a)&sup2; = (R: x⁴ + 4x&sup2;a + 4a&sup2;)
h) (x - 5) (x + 5) = (R: x&sup2; - 25)
i) (9y + 4 ) (9y - 4) = (R:81y&sup2; -16)
j) (m - n)&sup2; = (R: m&sup2; - 2mn + n&sup2;)
5) Sabendo que x&sup2; + y&sup2; = 153 e que xy = 36, calcule o valor de (x+y)&sup2;.
(R: 235)
6) Qual o valor num&eacute;rico da express&atilde;o (a - 2b)&sup2;, sabendo-se que a&sup2; + 4b&sup2; = 30 e ab = 5.
(R: 10)
7) Simplifique as express&otilde;es:
a) (x+y)2–x2-y2
(x+y)2–x2-y2 = x2+2xy+y2–x2-y2 = 2xy
b) (x+2)(x-7)+(x-5)(x+3)
(x+2)(x-7)+(x-5)(x+3) = x2+(2+(-7))x+2.(-7) + x2+(-5+3)x+3.(-5) =
x2-5x-14+ x2-2x-15 = 2x2-7x-29
c) (2x-y)2-4x(x-y)
(2x-y)2-4x(x-y) = (2x)2-2.2x.y+y2-4x2+4xy = 4x2-4xy+y2-4x2+4xy = y2
8) Desenvolva:
a) (3x+y)2
(3x+y)2 = (3x)2+2.3x.y+y2 = 9x2+6xy+y2
b) ((1/2)+x2)2
((1/2)+x2)2 = (1/2)2+2.(1/2).x2+(x2)2 = (1/4) +x2+x4
c) ((2x/3)+4y3)2
((2x/3)+4y3)2 = (2x/3)2-2.(2x/3).4y3+(4y3)2= (4/9)x2-(16/3)xy3+16y6
d) (2x+3y)3
(2x+3y)3 = (2x)3+3.(2x)2.3y+3.2x.(3y)2+(3y)3 = 8x3+36x2y+54xy2+27y3
e) (x4+(1/x2))3
(x4+(1/x2))3 = (x4)3+3.(x4)2.(1/x2)+3.x4.(1/x2)2+(1/x2)3 = x12+3x6+3+(1/x6)
f) ((2x/3)+(4y/5)).((2x/3)-(4y/5)
(2x/3)+(4y/5)).((2x/3)-(4y/5)) = (2x/3)2-(4y/5)2 = (4/9)x2-(16/25)y2
9) Se x - y = 7 e xy = 60, ent&atilde;o o valor da express&atilde;o x&sup2; + y&sup2; &eacute;:
a) 53
b) 109
c) 169
d) 420
Solu&ccedil;&atilde;o:
Do problema, temos a seguinte equa&ccedil;&atilde;o x - y = 7, a princ&iacute;pio n&atilde;o est&aacute; muito claro o valor de x&sup2; + y&sup2;, mas vamos tra&ccedil;ar
uma estrat&eacute;gia para resolu&ccedil;&atilde;o da quest&atilde;o:
Na equa&ccedil;&atilde;o x - y = 7, vamos elevar os dois membros ao quadrado, ficando assim:
(x - y)&sup2; = 7&sup2;, desenvolvendo temos:
x&sup2; - 2xy + y&sup2; = 49, veja que j&aacute; apareceram o x&sup2; e y&sup2;, arrumando
x&sup2; + y&sup2; = 49 + 2xy, mas xy = 60 e da&iacute;
x&sup2; + y&sup2; = 49 + 2.60, resolvendo:
x&sup2; + y&sup2; = 49 + 120, logo x&sup2; + y&sup2; = 169.
Utilizamos a estrat&eacute;gia de elevar os dois membros da equa&ccedil;&atilde;o ao quadrado - podemos fazer isto, desde que fa&ccedil;amos
em ambos os membros - e logo apareceu x&sup2; + y&sup2;.
10)A express&atilde;o (x - y)&sup2; - (x + y)&sup2; &eacute; equivalente a:
a) 0
b) 2y&sup2;
c) -2y&sup3;
d) -4xy
Solu&ccedil;&atilde;o:
Primeiro vamos desenvolver os bin&ocirc;mios separadamente:
(x - y)&sup2; - (x + y)&sup2;
(x-y)&sup2; = x&sup2; - 2xy + y&sup2; e (x + y)&sup2; = x&sup2; + 2xy + y&sup2;
Ap&oacute;s desenvolver, voltamos para a express&atilde;o e substitu&iacute;mos:
(x - y)&sup2; - (x + y)&sup2; = x&sup2; - 2xy + y&sup2; - (x&sup2; + 2xy + y&sup2;) = x&sup2; - 2xy + y&sup2; - x&sup2; - 2xy - y&sup2; =
x&sup2; - x&sup2; - 2xy - 2xy + y&sup2; - y&sup2; = -2xy - 2xy = - 4xy
Logo, (x-y)&sup2; - (x + y)&sup2; = - 4xy
11) (TRT-2011) Indagado sobre o n&uacute;mero de processos que havia arquivado certo dia, um T&eacute;cnico Judici&aacute;rio, que
gostava muito de Matem&aacute;tica, respondeu:
- O n&uacute;mero de processos que arquivei &eacute; igual a (12,25)2 - (10,25)2
Chamando X o total de processos que ele arquivou, ent&atilde;o &eacute; correto afirmar que:
a)38 &lt; X &lt; 42.
b) X &gt; 42.
c) X &lt; 20.
d)20 &lt; X &lt; 30.
e)30 &lt; X &lt; 38
Solu&ccedil;&atilde;o:
Temos que o produto da soma pela diferen&ccedil;a de dois termos pode ser vista como:
12) Calcule o produto da soma pela diferen&ccedil;a de dois termos:
a) (x + y) . ( x - y) =
b) (y – 7 ) . (y + 7) =
c) (x + 3) . (x – 3) =
d) (2x + 5 ) . (2x – 5) =
e) (3x – 2 ) . ( 3x + 2) =
f) (5x + 4 ) . (5x – 4) =
g) (3x + y ) (3x – y) =
h) ( 1 – 5x) . (1 + 5x) =
i) (2x + 3y) . (2x – 3y) =
j) (7 – 6x) . ( 7 + 6x) =
l) (1 + 7x&sup2;) . ( 1 – 7x&sup2;) =
13) Desenvolva:
a) ( x + y)&sup3; =
b) (x – y)&sup3; =
c) (m + 3)&sup3; =
d) (a – 1 )&sup3; =
e) ( 5 – x)&sup3; =
14) A express&atilde;o (a + b + c)&sup2; &eacute; igual a
a) a&sup2; + 2ab + b&sup2; + c&sup2;
b) a&sup2; + b&sup2; + c&sup2; + 2ab + 2ac + 2bc
c) a&sup2; + b&sup2; + c&sup2; + 2abc
d) a&sup2; + b&sup2; + c&sup2; + 4abc
e) a&sup2; + 2ab + b&sup2; + 2bc + c&sup2;
15) Seja N o resultado da opera&ccedil;&atilde;o 375&sup2; - 374&sup2;. A soma dos algarismos de N &eacute;:
a) 18
b) 19
c) 20
d) 21
e) 22
16) Efetuando-se (579865)&sup2; - (579863)&sup2;, obt&eacute;m-se
a) 4
b) 2 319 456
c) 2 319 448
d) 2 086 246
e) 1 159 728
17) O produto (x + 1)(x&sup2; - x +1) &eacute; igual a:
a) x&sup3;-1
b) x&sup3; + 3x&sup2; - 3x + 1
c) x&sup3; + 1
d) x&sup3; - 3x&sup2; + 3x - 1
e) x&sup2; + 2
Gabarito:
12) a) (R : x&sup2; - y&sup2;) b) ( R : y&sup2; - 49) c) ( R: x&sup2; - 9) d) ( R: 4x&sup2; - 25) e) ( R: 9x&sup2; - 4 )
f) ( R: 25x&sup2; - 16) g) (R: 9x&sup2; - y&sup2; ) h) ( R: 1 - 25x&sup2; ) i) ( R: 4x&sup2; - 9y&sup2; ) j) (R: 49 - 36x&sup2;) l) (R: 1 - 49x⁴)
13) a) (R: x&sup3; + 3x&sup2;y + 3xy&sup2; + y&sup3;) b) (R: x&sup3; - 3x&sup2;y + 3xy&sup2; - y&sup3;) c) ( R: m&sup3; + 9m&sup2; + 27m +27)
d) (R: a&sup3; - 3a&sup2; + 3a -1) e) (R: 125 - 75x + 15x&sup2; -x&sup3;)
14) B
15) C
16) B
17) C
Fatora&ccedil;&atilde;o de polin&ocirc;mios
1) Fatore o polin&ocirc;mio ax&sup2; + bx&sup2; - 7x&sup2;.
x&sup2; .( a + b - 7 )
2) Escreva a forma fatorada do polin&ocirc;mio 8a5 b + 12a&sup3;.
4a&sup3;. ( 2a&sup2;b + 3 )
3) Fatore os seguintes polin&ocirc;mios:
a) 5x + 5y
5. ( x + y )
b) 7ab – 14bx
7b. ( a - 2x )
c) a&sup3; + 3a&sup2; + 5a
a.( a&sup2; + 3a + 5 )
d) 4x&sup2; + 12x&sup3;y – 28x&sup2;z
4x&sup2;. ( 1 + 3xy - 7z )
4) Fatore o polin&ocirc;mio 21a&sup2;b&sup2;c&sup3; + 9abc – 6abcd.
3abc. ( 7abc&sup2; + 3 - 2d )
5) Qual &eacute; o valor num&eacute;rico do polin&ocirc;mio 2m + 2n , sabendo que m + n = 10?
2( m + n ) =
2. 10 = 20
6) Que valor num&eacute;rico tem a express&atilde;o 5ab + 5a&sup2;, quando a = 4 e a + b = 8?
5a.( b + a )
5.4.8
20.8 = 160
7) Fatore :
a) 3a + 6b
3.( a + 2b )
b) 4x + 8
4. ( x + 2 )
c) – 2a – 4b
- 2.( a + 2b )
d) – 10m – 5n
- 5.( 2m + n )
8) Fatore as express&otilde;es.
a) x&sup2; - 4
(x - 2 ). ( x + 2 )
c) 4a&sup2; - 9b&sup2;
b) a&sup2; - 1
(a-1)(a+1)
d) 9x4 – 16y6 (3x&sup2; - 4y&sup3; ). (3x&sup2; + 4y&sup3;)
( 2a - 3b ) ( 2a + 3b )
a) x&sup2; + 12x + 64
n&atilde;o
e) a&sup2;x&sup2; + 2ax + 1
sim
b) a&sup2; - 22a + 121
sim
f) y&sup2; - 2y + 4
n&atilde;o
c) 4b&sup2; + 10b + 25
n&atilde;o
g) x&sup2; + 5x + 16
n&atilde;o
a) x4 + 8x&sup2; + 16
( x&sup2; + 4)&sup2;
c) m&sup2; - 6mn + 9n&sup2;
( m - 3n)&sup2;
b) 1 + 2x&sup2;y&sup3; + x4y4
( 1 + x&sup2;y&sup2; )&sup2;
d) 1/4a2b&sup2; - 5a&sup2;b + 25a&sup2; ( 1/2ab - 5a)&sup2;
11) De a forma fatorada das seguintes express&otilde;es.
a) 2x + 2y + 3x + 3y
c) 3a – 3b + ma – mb
5.( x + y )
(3+m)(a-b)
b) a – ax + b – bx + c – cx
(a+b+c)(1-x)
12) Qual o valor num&eacute;rico da express&atilde;o ax + ay + 3x + 3y, sabendo-se que
a = 2 e x + y = 5? 25
13) Express&otilde;es alg&eacute;bricas fatoradas (fatora&ccedil;&atilde;o simples).
a) ax + ay + az = a(x + y + z)
b) 4m2 + 6am =2m(2m 3a)
c) 7xy2 - 21x2y = 7xy(y - 3x)
14) Express&otilde;es alg&eacute;bricas fatoradas (por agrupamento)
a) ax + bx + am + bm = x(a + b) + m(a + b) = (a + b).(x + m)
b) 2x + 4y + mx + 2my = 2(x + 2y) + m(x + 2y) = (x + 2y).(2 + m)
a) 9x2 - 16 = (3x - 4).(3x + 4)
b) 25 - 4a2m6 = (5 - 2am3).(5 + 2m3)
c) 0, 81b4 - 36 = (0,9b2 - 6).(0,9b2 + 6)
d) (a + 3)2 - 9 = (a + 3 - 3).(a + 3 +3) = a(a + 6)
e) (m + 1)2 - (k - 2)2 = [(m + 1 - (k - 2].[m + 1 + (k -2)] =
(m +1 - k + 2).(m + 1 + k - 2) = (m - k +3).(m + k - 1)
a) x2 - 4x + 4 = (x - 2)2 = (x - 2).(x - 2)
b) x2 - 6x + 9 = (x - 3)2 = (x - ).(x - 3)
c) x2 - 10x + 25 = (x - 5)2 = (x - 5).(x - 5)
d) m2 + 8m + 16 = (m + 4)2 = (m + 4).(m + 4)
e) p2 - 2p + 1 = (p - 1)2 = (p - 1).(p - 1)
f) k4 + 14k2 + 49 = (k2 + 7)2 = (k2 + 7).(k2 + 7)
g) (m + 1)2 - 6(m + 1) + 9 = (m + 1 - 3)2 = (m - 2)2 = (m - 2).(m - 2)
16) Fatora&ccedil;&atilde;o da soma e da diferen&ccedil;a de dois cubos
a) a3 + b3 = (a + b).(a2 - ab + b2)
b) m3 - 8n3 = m3 - (2n)3 = (m - 2n)(m2 + 2mn + 4n2)
c) x6 + 64 = (x2)3 + 43 = (x2 + 4).(x4 - 4x2 + 16)
d) y3 - 125 = y3 - 53 = (y - 5).(y2 + 5y + 25)
17) Fatore at&eacute; as express&otilde;es tornarem-se irredut&iacute;veis:
a) m8 - 1 = (m4)2 - 1=
(m4 + 1) (m4 - 1). =
(m4 + 1) (m2 + 1) (m2 - 1)(m2 + 1)=
(m4 + 1) (m2 + 1) (m + 1).(m – 1).
b) ax3 - 10ax2 + 25ax = ax(x2 - 10x + 25) = ax(x - 5)2 = ax(x - 5).(x - 5)
c) 2m3 - 18m = 2m(m2 - 9) = 2m(m - 3).(m + 3)
d) [(x -3)2 - 4(x - 3) + 4] - [(x - 3)2 + 4(x - 3) + 4] = [(x - 3 - 2)2] - [(x - 3 + 2)2 = (x - 5)2 - (x - 1)2
[(x - 5 - (x - 1)] .[x - 5 + (x - 1)] = - 4(2x - 6) = - 4. 2(x - 3) = -8.(x - 3)
```