Polynomials practice

Remainder theorem worksheets

Use the Remainder Theorem in twelve structured problems: evaluate four polynomials at a point, make four factor decisions from zero remainders, and divide four polynomials by signed monic linear divisors with exact quotient-and-remainder answers.

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Evaluate P(c), decide whether x − c is a factor, then divide and state the quotient and remainder.

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One theorem, three connected tasks

Evaluating P(c) gives the same number as the remainder after division by x − c. A zero value proves that x − c is a factor; a nonzero value is the signed remainder.

Write a signed divisor as x − c before substituting. For example, x + 3 means c = −3. Check every division with P(x) = (x − c)Q(x) + R.

Learning objective

What this activity builds

Connect P(c), divisibility by x − c, and the quotient-remainder identity for polynomial division.

Worked example

For P(x) = x³ − 2x² − 5x + 6, evaluate P(1) and decide whether x − 1 is a factor.

P(1) = 0, so x − 1 is a factor.

The Remainder Theorem says that the remainder after division by x − c equals P(c). A zero remainder makes the division exact.

Teaching notes

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Substitute the entire signed value of c into every occurrence of the variable.
Match x − c carefully: x + 3 corresponds to c = −3.
A divisor is a factor exactly when the remainder is zero.
Check division by confirming P(x) = (x − c)Q(x) + R.

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Dividing polynomials worksheets

Divide polynomials by monic linear divisors in ten expression problems or six long-division problems. Choose binomial, trinomial, four-term or mixed dividends and check exact quotient-and-remainder keys.

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