Question

Temos as funções:
f(x) = x+1
g(x)= x³+ax²+bx+c
h(x)= g(f(x))
Considerando as raízes de h(x) são {-1;0;1} determine h(-2)

Answer 1

Boa noite Claudia

f(x) = x + 1
g(x) = x³ + ax² + bx + c 
h(x) = g(f(x))

h(x) = (x + 1)³ + a*(x + 1)² + b*(x + 1) + c
h(x) = x³ + 3x² + 3x + 1 + ax² + 2ax + a + bx + b + c
h(x) = x³ + (a + 3)x²  + (3 + 2a + b)x + 1 + a + b + c

h(-1) = -1 + a + 3 - 3 - 2a - b + 1 + a + b + c = 0
c = 0

h(0) = 0 + 0 + 0 + 1 + a + b + 0 = 0
a + b = -1 

h(1) = 1 + a + 3 + 3 + 2a + b + 1 + a + b + 0 = 0
4a + 2b = -8

a + b = -1
2a + b = -4
-a = 3
a = -3
-3 + b = -1
b = 2

h(x) = x³ + (-3 + 3)x²  + (3 - 6 + 2)x + 1 - 3 + 2 + 0
h(x) = x³ - x 

h(-2) = (-2)³ - (-2) = -8 + 2 = -6