Remainder and factor theorems

Japanese school year: Math II

What you learn

You will learn the remainder theorem for finding polynomial remainders quickly, along with the factor theorem for identifying linear factors. These theorems are powerful tools for solving higher-degree equations and factoring polynomials. Familiarity with polynomial long division and evaluating algebraic expressions is recommended beforehand.

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Key points

This theorem states that the remainder when dividing polynomial $P(x)$ by $x - \alpha$ equals $P(\alpha)$ . It lets you find the remainder quickly without long division.

P(x) = (x - \alpha)Q(x) + P(\alpha)

When dividing $P(x)$ by $ax + b$ , the remainder is $P(-b/a)$ , obtained by setting $ax + b = 0$ . Use this when dividing by a linear factor with coefficient $a$ .

P(x) = (ax + b)Q(x) + P\left(-\frac{b}{a}\right)

If $P(\alpha) = 0$ , polynomial $P(x)$ is divisible by $x - \alpha$ and has $(x - \alpha)$ as a factor. This helps factor higher-degree polynomials and solve equations.

P(\alpha) = 0 \iff P(x) = (x - \alpha)Q(x)

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Practice 1Coming soon