Question

Find values of a and b if (x² +1) is a factor of the polynomial x⁴ + x³ + 8x² + ax + b.

Answer 1

If f(x) is divisible by g(x) then  remainder will be zero. So to find the values of a and b, find the remainder and put it equal to zero to get the values of a and b.

SOLUTION:
Given: p(x) = x⁴  +  x³ +  8x²  +  ax +  b and g(x)= x² +1.

On dividing p(x) by g(x) the division process IS IN THE ATTACHMENT.
The Quotient = x²+x+7 and Remainder = x(a-1)+(b-7)

Since,  x⁴  +  x³ +  8x²  +  ax +  b is exactly divisible by x²+1, therefore the remainder should be zero.
So put x(a-1)+(b-7) = 0
x(a-1)+(b-7) = 0.x +0
a-1= 0 & b-7= 0
[On comparing the Coefficients of x and constant terms]
a= 1 & b= 7

Hence, the values of a = 1 &  b = 7.

HOPE THIS WILL HELP YOU...

Answer 2

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