Question

Find values of a and b so that x^{​​​​​​4} + x^{​​​​​​3} + 8x² + ax + b is divisible by x^{​​​​​​2} + 1.

Answer 1

Given polynomial is x4 + x3 + 8x2 + ax + b

Since x2 + 1 divides x4 + x3 + 8x2 + ax + b , so the quotient will be a polynomial of degree 2.

So, we can write

x4 + x3 + 8x2 + ax + b = (x2 + 1) (a1x2 + b1x + c1)

⇒ x4 + x3 + 8x2 + ax + b = a1x4 + a1x2 + b1x3 + b1x + c1x2 + c1

⇒ x4 + x3 + 8x2 + ax + b = a1x4 + b1x3 + (a1 + c1) x2 + b1x + c1

Comparing the coefficient of x4 on both sides, we get –

a1 = 1

On comparing the coefficient of x3, we get –

b1 = 1

On comparing the coefficient of x2, we get –

a1 + c1 = 8

⇒ 1 + c1 = 8

⇒ c1 = 7

On comparing the coefficient of x on both sides, we get –

a = b1 = 1

⇒ a = 1

On comparing the constants on both sides, we get –

b = c1 = 7

⇒ b = 7

Hence, values of a and b are 1 and 7.