Factoring Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factoring.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Factoring reveals structure—roots, common factors, and simplifications.

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

Sense of Study hint: Write out all factor pairs of the constant term and test which pair sums to the middle coefficient.

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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Factor x^2 - 16.

Example 2

medium
Factor 3x^2 - 12x.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

polynomialsmultiplication