Question 1

By the Factor Theorem, $(x - 2)$ is a factor of $p(x)$ exactly when:

Accepted Answer

p(2) = 0 — $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$; here $a = 2$.

Question 2

By the Remainder Theorem, dividing $p(x)$ by $(x - 3)$ leaves a remainder equal to:

Accepted Answer

p(3) — The remainder when $p(x)$ is divided by $(x-a)$ is always $p(a)$; here $a=3$.

Question 3

For $p(x) = x^3 - 2x^2 - 5x + 6$, verify that $x = 1$ is a root by finding $p(1)$.

Accepted Answer

0 — $p(1) = 1 - 2 - 5 + 6 = 0$, confirming $x = 1$ is a root and $(x-1)$ is a factor.

Polynomials

Sample practice questions