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2. POLINÔMIOS
1
01. (Efomm 2020) Considere a inequação ( )
7 4 2 2
x x x 1 x 4x 3 x 7x 54 0.
− + − − + − − Seja I o conjunto dos números
inteiros que satisfaz a desigualdade e n a quantidade de elementos de I. Com relação a n, podemos afirmar que
a) n é um número primo.
b) n é divisível por 7.
c) n não divide 53904.
d) n é um quadrado perfeito.
e) n é divisível por 6.
02. (Ita 2020) Seja 4 3 2
p(x) ax bx cx dx e
= + + + + um polinômio com coeficientes reais. Sabendo que:
I. p(x) é divisível por 2
x 4;
−
II. a soma das raízes de p(x) é igual a 1;
III. o produto das raízes de p(x) é igual a 3;
IV.
15
p( 1) ;
4
− = −
então, p(1) é igual a
a)
17
.
2
− b)
19
.
4
− c)
3
.
2
− d)
9
.
4
e)
9
.
2
03. (Espcex 2020) Dividindo-se o polinômio 4 3
P(x) 2x 5x kx 1
= − + − por (x 3)
− e (x 2),
+ os restos são iguais. Neste
caso, o valor de k é igual a
a) 10.
b) 9.
c) 8.
d) 7.
e) 6.
04. (Ime 2020) Um polinômio P(x) de grau maior que 3 quando dividido por x 2, x 3
− − e x 5
− deixa restos 2, 3 e
5, respectivamente. O resto da divisão de P(x) por (x 2)(x 3)(x 5)
− − − é
a) 1
b) x
c) 30
d) x 1
−
e) x 30
−
05. (Ita 2020) Considere o polinômio 3 2
p(x) x mx x 5 n,
= − + + + sendo m, n números reais fixados. Sabe-se que toda
raiz z a bi,
= + com 𝑎, 𝑏 ∈ ℝ, da equação p(z) 0
= satisfaz a igualdade 2
a mb nb 1.
= + − Então, a soma dos quadrados
das raízes de p(z) 0
= é igual a
a) 6.
b) 7.
c) 8.
d) 9.
e) 10.
3. POLINÔMIOS
2
06. (Epcar 2020) Considere os polinômios na variável x :
3 3 2
A(x) x (3m 4m)x 2,
= + − − sendo 𝑚 ∈ ℚ; e
2
B(x) x 2x 1
= − +
Os gráficos de A(x) e B(x) possuem apenas um ponto comum sobre o eixo das abscissas.
É correto afirmar que
a) o produto e a soma das raízes imaginárias de A(x) são números conjugados.
b) os afixos das raízes de A(x) formam um triângulo equilátero.
c) as raízes de A(x) possuem argumentos que NÃO formam uma Progressão Aritmética.
d) todas as raízes de A(x) possuem o mesmo módulo.
07. (Espcex 2020) Se a equação polinomial 2
x 2x 8 0
+ + = tem raízesa e b e a equação 2
x mx n 0
+ + = tem raízes
(a 1)
+ e (b 1),
+ centro m n
+ é igual a
a) 2.
−
b) 1.
−
c) 4.
d) 7.
e) 8.
08. (Espcex 2020) Sabe-se que as raízes da equação 3 2
x 3x 6x k 0
− − + = estão em progressão aritmética. Então
podemos afirmar que o valor de
k
2
é igual a
a)
5
.
2
b) 4.
c)
7
.
2
d) 3.
e)
9
.
2
09. (Acafe 2019) Analise as afirmações a seguir e assinale a alternativa correta.
a) A equação 3 2
x 2x 3 0
+ + = possui pelo menos uma raiz irracional.
b) O resto da divisão de 15 4
p(x) x 3x 2x 3
= − + + por q(x) x 1
= + é 3.
c) Se 3 2
p(x) x 5x ax b
= + + + é divisível por x 1
+ e o quociente dessa divisão é um polinômio com raiz dupla então a
e b são primos entre si.
d) Se 3 2 2
2x A Bx C
,
x 2
x 2x 4x 8 x 4
+
= +
−
− + − +
então A B C 1.
+ − =
4. POLINÔMIOS
3
10. (Espcex 2019) Sabendo que o número complexo i (sendo i a unidade imaginária) é raiz do polinômio
5 4
p(x) x 2x x 2,
= − − + podemos afirmar que p(x) tem
a) duas raízes iguais a i, uma raiz racional e duas raízes irracionais.
b) i e i
− como raízes complexas e três raízes irracionais.
c) uma raiz complexa i e quatro raízes reais.
d) i e i
− como raízes complexas e três raízes inteiras.
e) três raízes simples e uma raiz dupla.
11. (Ita 2019) Seja 3 2
p(x) x ax bx
= + + um polinômio cujas raízes são não negativas e estão em progressão aritmética.
Sabendo que a soma de seus coeficientes é igual a 10, podemos afirmar que a soma das raízes de p(x) é igual a
a) 9.
b) 8.
c) 3.
d)
9
.
2
e) 10.
12. (Ita 2019) Considere as seguintes afirmações:
I. se 1 2
x , x e 3
x são as raízes da equação 3 2
x 2x x 2 0,
− + + = então 1 2 3 2 1 3
y x x , y x x
= = e 3 1 2
y x x
= são as raízes da
equação 3 2
y y 4y 4 0.
− − − =
II. a soma dos cubos de três números inteiros consecutivos é divisível por 9.
III.
3 5 1 5
.
2 2
+ +
=
É(são) VERDADEIRA(S)
a) apenas I
b) apenas II
c) apenas III
d) apenas II e III
e) todas
13. (Ime 2019) Sejam 1 2
x , x e 3
x raízes da equação 3
x ax 16 0.
− − = Sendo a um número real, o valor de
3 3 3
1 2 3
x x x
+ + é igual a
a) 32 a
−
b) 48 2a
−
c) 48
d) 48 2a
+
e) 32 a
+
5. POLINÔMIOS
4
14. (Acafe 2018) Considere as funções f(x) 4
= e 3 2
g(x) x 3x .
= − + Os pontos A e B são as intersecções do gráfico
da função g com o eixo das abscissas. Os pontos G e H são as intersecções dos gráficos das funções f e g. O
quadrilátero de vértices ABGH tem área igual a
a) 6 u.a.
b) 4 u.a.
c) 12 u.a.
d) 18 u.a.
15. (Ime 2017) Sejam x, y e z números complexos que satisfazem ao sistema de equações abaixo:
2 2 2
x y z 7
x y z 25
1 1 1 1
x y z 4
+ + =
+ + =
+ + =
O valor da soma 3 3 3
x y z
+ + é
a) 210
b) 235
c) 250
d) 320
e) 325
16. (Ime 2017) O polinômio 3
P(x) x bx 80x c
= − + − possui três raízes inteiras positivas distintas. Sabe-se que duas
das raízes do polinômio são divisoras de 80 e que o produto dos divisores positivos de c menores do que c é 2
c .
Qual é o valor de b?
a) 11
b) 13
c) 17
d) 23
e) 29
17. (Esc. Naval 2017) Seja 6 5 4 3 2
P(x) x bx cx dx ex fx g
= + + + + + + um polinômio de coeficientes inteiros e que
3
P( 2 3) 0.
+ = O polinômio R(x) é o resto da divisão de P(x) por 3
x 3x 1.
− − Determine a soma dos coeficientes de
R(x) e assinale a opção correta.
a) 51
−
b) 52
−
c) 53
−
d) 54
−
e) 55
−
6. POLINÔMIOS
5
18. (Epcar 2017) Sejam Q(x) e R(x) o quociente e o resto, respectivamente, da divisão do polinômio 3 2
x 6x 9x 3
− + −
pelo polinômio 2
x 5x 6,
− + em que 𝑥 ∈ ℝ. O gráfico que melhor representa a função real definida por
P(x) Q(x) R(x)
= + é
a) b) c) d)
19. (Eear 2017) Considere 3 2
P(x) 2x bx cx,
= + + tal que P(1) 2
= − e P(2) 6.
= Assim, os valores de b e c são,
respectivamente,
a) 1 e 2
b) 1 e 2
−
c) 1
− e 3
d) 1
− e 3
−
20. (Acafe 2017) Seja P(x) um polinômio divisível por (x 2).
− Se dividirmos o polinômio P(x) por 2
(x 2x),
+ obteremos
como quociente o polinômio 2
(x 2)
− e resto igual a R(x). Se R(3) 6,
= então, a soma de todos os coeficientes de P(x)
é igual a
a) 38.
−
b) 41.
−
c) 91.
d) 79.
GABARITO
1 - D 2 - D 3 - B 4 - B 5 - B
6 - C 7 - D 8 - B 9 - A 10 - D
11 - A 12 - E 13 - C 14 - C 15 - B
16 - E 17 - E 18 - A 19 - D 20 - B