find the value of a and b so that x4+x3+8x2+ax+b is divisible by x2+1
Given polynomial is x 4 + x 3 + 8x 2 + ax + b
Since x 2 + 1 divides x 4 + x 3 + 8x 2 + ax + b , so the quotient will be a polynomial of degree 2.
So, we can write
x 4 + x 3 + 8x 2 + ax + b = (x 2 + 1) (a 1 x 2 + b 1 x + c 1)
⇒ x 4 + x 3 + 8x 2 + ax + b = a 1 x 4 + a 1 x 2 + b 1 x 3 + b 1 x + c 1 x 2 + c 1
⇒ x 4 + x 3 + 8x 2 + ax + b = a 1 x 4 + b 1 x 3 + (a 1 + c 1) x 2 + b 1 x + c 1
Comparing the coefficient of x 4 on both sides, we get –
a 1 = 1
On comparing the coefficient of x 3, we get –
b 1 = 1
On comparing the coefficient of x 2, we get –
a 1 + c 1 = 8
⇒ 1 + c 1 = 8
⇒ c 1 = 7
On comparing the coefficient of x on both sides, we get –
a = b 1 = 1
⇒ a = 1
On comparing the constants on both sides, we get –
b = c 1 = 7
⇒ b = 7
Hence, values of a and b are 1 and 7.