Grades Algebra II & Trigonometry Polynomials & Rational ExpressionsPolynomial Division & the Remainder Theorem

Polynomial Division & the Remainder Theorem

📐 Polynomials & Rational Expressions

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Polynomial Division & the Remainder Theorem

Divide polynomials and apply the Remainder Theorem to evaluate and factor.

Polynomial long division generalizes numeric long division. When dividing p(x) by divisor d(x), we get p(x) = d(x) · q(x) + r(x), where the degree of r is less than the degree of d. Synthetic division is a streamlined version when dividing by (x − c).

p(x) = d(x) \cdot q(x) + r(x) \qquad \text{Remainder Theorem: } p(c) = r \text{ when dividing by }(x-c)

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The Remainder Theorem: when p(x) is divided by (x − c), the remainder equals p(c). Consequently, (x − c) is a factor of p(x) if and only if p(c) = 0.

Synthetic Division of p(x) ÷ (x − c)

1Write c in a box; list the coefficients of p(x) horizontally.

2Bring down the first coefficient.

3Multiply by c and write the result under the next coefficient; add.

4Repeat until all coefficients are used.

5The last number is the remainder; the others are coefficients of q(x).

🔢Using the Remainder Theorem

Is (x − 2) a factor of p(x) = x³ − 4x² + x + 6? Compute p(2) = 8 − 16 + 2 + 6 = 0. Yes! Since p(2) = 0, (x − 2) is a factor.

p(x) = x^3 - 4x^2 + x + 6,\quad p(2) = 8 - 16 + 2 + 6 = 0 \implies (x-2) \mid p(x)

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Remember This!

Use the Rational Root Theorem to identify candidate rational roots: factors of the constant term divided by factors of the leading coefficient. Then test with synthetic division.

✏️ Try It!

What is the remainder when p(x) = 2x² + 5x + 1 is divided by (x − 2)?

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