Grades Algebra II & Trigonometry Polynomials & Rational ExpressionsFactoring Higher-Degree Polynomials

Factoring Higher-Degree Polynomials

📐 Polynomials & Rational Expressions

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Factoring Higher-Degree Polynomials

Factor cubic and higher-degree polynomials using multiple techniques.

Factoring higher-degree polynomials combines multiple strategies: pulling out the GCF, recognizing special patterns (sum/difference of cubes, grouping), and using roots found by the Rational Root Theorem paired with synthetic division.

a^3 + b^3 = (a+b)(a^2 - ab + b^2) \qquad a^3 - b^3 = (a-b)(a^2 + ab + b^2)

Factoring by Grouping

Group terms into pairs.Factor GCF from each pair.Factor out the common binomial factor.Example: x³+x²+3x+3 = x²(x+1)+3(x+1) = (x²+3)(x+1)

Sum/Difference of Cubes

a³ + b³ = (a+b)(a²−ab+b²)a³ − b³ = (a−b)(a²+ab+b²)x³ − 8 = (x−2)(x²+2x+4)27x³ + 1 = (3x+1)(9x²−3x+1)

Always try GCF first, then look for special patterns before attempting grouping or the Rational Root Theorem.

x^3 - 8 = (x-2)(x^2 + 2x + 4)

🔓Full Factoring Strategy

Factor x³ + 3x² − x − 3. Group: (x³ + 3x²) + (−x − 3) = x²(x+3) − 1(x+3) = (x² − 1)(x + 3) = (x+1)(x−1)(x+3).

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Remember This!

For a cubic with rational roots, once you find one root r (using the Rational Root Theorem and synthetic division), divide to get a quadratic, then factor the quadratic with standard methods.

✏️ Try It!

Which is the correct factorization of x³ − 27?

Take Quiz 📝 — 24 Questions