Factoring Formula

The Formula

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

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

Quick Example

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). 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), you compress x^2 + 5x + 6 into the same product.

Formal View

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

Worked Examples

Example 1

easy
Factor x^2 + 7x + 12.

Solution

  1. 1
    Find two numbers that multiply to 12 and add to 7: those are 3 and 4.
  2. 2
    Write the factored form: (x + 3)(x + 4).
  3. 3
    Check by expanding: x^2 + 4x + 3x + 12 = x^2 + 7x + 12 ✓

Answer

(x + 3)(x + 4)
To factor x^2 + bx + c, find two numbers p and q such that p + q = b and p \cdot q = c. Then the factorization is (x + p)(x + q).

Example 2

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

Common Mistakes

  • Missing negative factors
  • Not factoring completely

Why This Formula Matters

Key technique for solving equations and simplifying expressions.

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), you compress x^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). The original expression and its factored form are connected by =.

Why is the Factoring formula important in Math?

Key technique for solving equations and simplifying expressions.

What do students get wrong about Factoring?

Finding the right pair of numbers that multiply to ac and add to b takes practice; not every polynomial factors over integers.

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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