Question

On dividing a polynomial p(x) by a nonzero polynomial q(x), let g(x) be the quotient and r(x) be the remainder then p(x) = q(x) · g(x) + r(x), where

Options

• r(x) = 0 always

• deg r(x) < deg g(x) always

• either r(x) = 0 or deg r(x) < deg g(x)

• r(x) = g(x)

Answer 1

either r(x) = 0 or deg r(x) < deg g(x)

Explanation:

By the polynomial division algorithm, when a polynomial p(x) is divided by a nonzero polynomial g(x) the remainder r(x) must either be the zero polynomial (if g(x) divides p(x) exactly) or have degree strictly less than deg g(x); if deg r(x) were ≥ deg g(x) we could continue dividing to reduce it further.