Question

what must be added to the polynomial p(x)=x³ + 6x² + 11x + 18, so that the resulting polynomial is exactly divisible by x² + 2x + 3?

Answer 1

p(x) = x³+6x²+11x+18

Let q(x) = x²+2x+3

p(x)/q(x) \(=\frac{x^3+6x^2+11x+18}{x^2+2x+3}\)

= x + \(\frac{4x^2+8x+18}{x^2+2x+3}\)

= (x+4) + 6/x²+2x+3

∴ p(x) = (x+4) (x²+2x+3) + 6

⇒ p(x) - 6 = (x+4) (x²+2x+3)

Therefore, we must add -6 to p(x) so that the resulting polynomial is exactly divisible by q(x) = x²+2x+3.