Remainder and Factor Theorems

Algebra Foundation

Algebra 2

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Algebra 2 · Unit 3 · Topic 3.3

Remainder and Factor Theorems

When P(x) is divided by x − c, the remainder is P(c).

Definition and explanation

When P(x) is divided by x − c, the remainder is P(c). Consequently, x − c is a factor exactly when P(c) = 0.

Worked example

For P(x) = x³ − 4x + 2, division by x − 2 leaves remainder P(2) = 8 − 8 + 2 = 2. Therefore x − 2 is not a factor; no complete division is needed to decide this.

Common mistake

A small remainder is not a zero remainder: a factor requires exact equality to zero.

Key takeaway

When P(x) is divided by x − c, the remainder is P(c).

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.