Factoring Formula

Factoring is rewriting an algebraic expression as a product of two or more simpler expressions that multiply to give the original.

The Formula

Key patterns: a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b), a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a+b)^2, a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a-b)^2

When to use: Reverse distribution: instead of expanding (x+2)(x+3)(x+2)(x+3), you compress x2+5x+6x^2 + 5x + 6 into the same product.

Quick Example

x2−9=(x+3)(x−3)x^2 - 9 = (x + 3)(x - 3) — a difference of squares; verify by expanding to confirm.

Notation

Factored form uses parentheses for each factor: (x+a)(x+b)(x + a)(x + b). The original expression and its factored form are connected by ==.

What This Formula Means

Rewriting an algebraic expression as a product of two or more simpler expressions that multiply to give the original.

Reverse distribution: instead of expanding (x+2)(x+3)(x+2)(x+3), you compress x2+5x+6x^2 + 5x + 6 into the same product.

Formal View

Factoring a polynomial P(x)∈R[x]P(x) \in \mathbb{R}[x] means writing P(x)=an∏i=1k(x−ri)mi⋅Q(x)P(x) = a_n \prod_{i=1}^{k}(x - r_i)^{m_i} \cdot Q(x) where rir_i are real roots with multiplicities mim_i and Q(x)Q(x) is irreducible over R\mathbb{R}.

Worked Examples

Example 1

easy
Factor x2+7x+12x^2 + 7x + 12.

Answer

(x+3)(x+4)(x + 3)(x + 4)

First step

1
Find two numbers that multiply to 12 and add to 7: those are 3 and 4.

Full solution

  1. 2
    Write the factored form: (x+3)(x+4)(x + 3)(x + 4).
  2. 3
    Check by expanding: x2+4x+3x+12=x2+7x+12x^2 + 4x + 3x + 12 = x^2 + 7x + 12 ✓
To factor x2+bx+cx^2 + bx + c, find two numbers pp and qq such that p+q=bp + q = b and p⋅q=cp \cdot q = c. Then the factorization is (x+p)(x+q)(x + p)(x + q).

Example 2

medium
Factor 6x2+11x+36x^2 + 11x + 3.

Example 3

medium
Factor: x2+2x−15x^2 + 2x - 15.

Common Mistakes

  • Forgetting to pull out the greatest common factor first - factor out the GCF before any other pattern.
  • Sign errors in the binomials - for x2−5x+6x^2-5x+6 both factors are negative: (x−2)(x−3)(x-2)(x-3).
  • Assuming every quadratic factors over integers - if no integer pair works, switch to the quadratic formula.

Why This Formula Matters

A product equal to zero is solvable instantly via the zero-product property, which is why factoring underlies most quadratic solving. It also exposes common factors that cancel in rational expressions, simplifying work later. Recognizing it by "Am I rewriting an expression as a product of simpler factors that multiply back to it?" — rather than by familiar numbers — is what lets a student tell it apart from expanding/distributing and quadratic formula and simplifying in a mixed problem set.

Frequently Asked Questions

What is the Factoring formula?

Rewriting an algebraic expression as a product of two or more simpler expressions that multiply to give the original.

How do you use the Factoring formula?

Reverse distribution: instead of expanding (x+2)(x+3)(x+2)(x+3), you compress x2+5x+6x^2 + 5x + 6 into the same product.

What do the symbols mean in the Factoring formula?

Factored form uses parentheses for each factor: (x+a)(x+b)(x + a)(x + b). The original expression and its factored form are connected by ==.

Why is the Factoring formula important in Math?

A product equal to zero is solvable instantly via the zero-product property, which is why factoring underlies most quadratic solving. It also exposes common factors that cancel in rational expressions, simplifying work later. Recognizing it by "Am I rewriting an expression as a product of simpler factors that multiply back to it?" — rather than by familiar numbers — is what lets a student tell it apart from expanding/distributing and quadratic formula and simplifying in a mixed problem set.

What do students get wrong about Factoring?

The procedure for factoring is the easy part; the trap is forgetting to pull out the greatest common factor first. Asking "Am I rewriting an expression as a product of simpler factors that multiply back to it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Factoring formula?

Before studying the Factoring formula, you should understand: polynomials, multiplication.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Factoring Polynomials: All Methods Explained with Step-by-Step Examples →
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