The central idea of all factoring is to rewrite a mathematical object as a product. In arithmetic, factoring is rewriting an integer as the product of its component numbers, especially into prime integers. In algebra, factoring is the rewriting of a polynomial as a product of polynomials of lower order (lower variable exponent).
Factoring Out a Common Factor

In simplifying a fraction, one finds the common factor in the numerator and denominator of the fraction.
Factoring by Grouping

A factoring problem with four terms often is a grouping problem. The method is to factor in two stages.
Factoring a Trinomial

A trinomial of form ax^2+bx+c can be factored by finding factors of the product ac that sum to b.
Factoring Perfect Squares

If a and c are squared integers, the trinomial can be a perfect square.
Factoring the Difference of Two Squares

The difference of two squares asquared + bsquared always factors into the product the sum and difference of a and b.
Factoring the Sum or Difference of Two Cubes

There are simple formulas for factoring the sum or difference of two cubes that can be verified by multiplying out the factors.