Factorization rewrites an expression as a product of simpler factors. It is a core algebra skill used to simplify expressions, solve equations, and identify common structure. The right identity depends on the terms and signs in the expression, so first look for a common factor and then check whether a standard pattern applies.

Common factorization formulas

  • Common factor: ab + ac = a(b + c)
  • Difference of squares: a² − b² = (a − b)(a + b)
  • Perfect-square sum: a² + 2ab + b² = (a + b)²
  • Perfect-square difference: a² − 2ab + b² = (a − b)²
  • Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)
  • Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)

Worked examples

For 6x + 12, the common factor is 6, so the expression becomes 6(x + 2). For x² − 9, recognise 9 as 3² and use the difference of squares: (x − 3)(x + 3). For x² + 6x + 9, the expression matches x² + 2(x)(3) + 3², giving (x + 3)².

How to factor an expression

  1. Look for the greatest common factor in every term.
  2. Check the number of terms and the powers of the variables.
  3. Compare the expression with a known identity.
  4. Write the factors and preserve the original signs.
  5. Expand the result to verify that it reproduces the starting expression.

Common mistakes

Do not confuse a sum of squares with a difference of squares: a² + b² does not factor over the real numbers in the same way as a² − b². Also check the middle term in a perfect-square trinomial; its magnitude must be 2ab.

Frequently asked questions

Why is factorization useful?

It can simplify expressions and make polynomial equations easier to solve.

How can I check my factors?

Multiply them together and compare the expanded expression with the original.

Can every expression be factored using these identities?

No. These identities cover common patterns, but other expressions may need grouping, polynomial methods, or may not factor over the chosen number system.