Question

Find a and b if f (x)is exactly divisible by g(x) where f(x)=x4+x3+ax2+bx-8 and g(x)=x2+x-2

Answer 1

Answer:

a = 2

b = 4

Step-by-step explanation:

g(x) = x² + x - 2 = ( x + 2 ) ( x - 1 )

So:

f(x) is divisible by g(x)

<=> f(x) is divisible by x+2   AND   f(x) is divisible by x-1

<=> f(-2) = 0   AND    f(1) = 0

<=> (-2)⁴ + (-2)³ + a(-2)² + b(-2) - 8 = 0   AND   1 + 1 + a + b - 8 = 0

<=> 16 - 8 + 4a - 2b - 8 = 0   AND   a + b = 6

<=> 4a - 2b = 0   AND   a + b = 6

<=> 2a = b   AND   a + b = 6

=> a + 2a = 6  =>  3a = 6 => a = 2

and then b = 2a = 4.