Factoring rewrites a polynomial as a product of simpler polynomials. It is the reverse of expanding a product.

For example,

$$x^2+5x+6=(x+2)(x+3).$$

Multiplying the factors on the right recovers the original polynomial.

Common factors

A factor shared by every term can be taken outside the parentheses. For example,

$$6x^2+9x=3x(2x+3).$$

The greatest common factor is usually removed first because it simplifies the remaining expression.

Useful patterns

Some products occur often enough to recognize directly. The difference of two squares satisfies

$$a^2-b^2=(a-b)(a+b).$$

For example,

$$x^2-9=(x-3)(x+3).$$

A quadratic trinomial may also split into two linear factors. To factor

$$x^2+5x+6,$$

we look for two numbers whose product is $6$ and whose sum is $5$. The numbers $2$ and $3$ give

$$x^2+5x+6=(x+2)(x+3).$$

A proposed factorization can always be checked by expanding the product again.