Question

Find the roots of ax² + bx + 6, if the polynomial x⁴ + x³ + 8x² + ax + b is exactly divisible by x² + 1.

(A) -1,3

(B) 2, 5

(C) -1, -6

(D) -3, 2

Answer 1

The correct option is: (C) -1, -6

Explanation:

x⁴ + x³ + 8x² + ax + b is exactly divisible by x² + 1

=> Remainder must be zero.

(a - 1)x + (b - 7) = 0

=> a - 1 = 0 and b - 7 = 0 => a = 1 and b = 7

Now, ax² + bx + 6 becomes x² + 7x + 6.

x² + 7x + 6 = x² + 6x + x + 6 = 0

= x(x + 6)+1(x + 6) = 0

= (x + 1)(x + 6) = 0 => x = -1,-6