Factoring Rules

High School

Definition

Rules that tell you how to properly handle different types of formulas when you're carrying out factoring. Examples of factoring rules include difference of squares, difference of cubes, and sum of squares. By observing the question you're dealing with, you can refer to the below rules to see how you can factor your formula.

Worked examples

\(x^2 - 9 = (x - 3)(x + 3)\)

Difference of squares: \(a^2 - b^2 = (a - b)(a + b)\), so \(x^2 - 3^2\) factors into two binomials.

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

Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) with \(a = x\) and \(b = 2\).

\(x^3 + 27 = (x + 3)(x^2 - 3x + 9)\)

Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) with \(a = x\) and \(b = 3\).

Common mistakes

\(x^2 + 9 = (x + 3)(x + 3)\) → \(x^2 + 9\) does not factor over real numbers Sum of squares cannot be factored using real numbers; only difference of squares factors.
\(x^3 + 8 = (x + 2)(x^2 + 2x + 4)\) → \(x^3 + 8 = (x + 2)(x^2 - 2x + 4)\) Sum of cubes has a minus sign in the middle term of the trinomial: \(a^2 - ab + b^2\).
\(x^3 - 27 = (x - 3)(x^2 + 9)\) → \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\) Difference of cubes needs the middle term \(ab\): \(a^2 + ab + b^2\), not just \(a^2 + b^2\).

Where you'll use it next

You'll apply factoring rules to solve polynomial equations, simplify rational expressions, and sketch graphs in algebra 2 and precalculus. They also appear in calculus when integrating rational functions.

Found in 1 StudyPug lesson

Factoring Trinomials: Mastering x^2 + bx + c

Algebra 1

Unlock the secrets of factoring trinomials with our comprehensive guide. Learn powerful techniques like decomposition and cross-multiplication to solve x^2 + bx + c effortlessly.

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