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

(A) -1,3

(B) 2, 5

(C) -1, -6

(D) -3, 2

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The correct option is: (C) -1, -6

Explanation:

x+ 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 = x+ 6x + x + 6 = 0

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

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

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