Question

Find the remainder when x4 - 5x3 + x2 - 10x - 5 is divided by x + 3.

Answer 1

Use the remainder theorem: we can decompose the given polynomial in terms of quotient and remainder polynomials $q(x)$ and $r(x)$, respectively, such that

$x^4 - 5x^3 + x^2 - 10x - 5 = (x + 3) q(x) + r(x)$

Then letting x = -3 makes the quotient term vanish, and we're left with a remainder of

$r(-3) = (-3)^4 - 5\times(-3)^4 + (-3)^2 - 10\times(-3) - 5 = \boxed{250}$