Find the values of a and b so that x4+x3+8x2+ax +b is divisible by x2+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.