Factoring Out a Common Factor

A polynomial written as a sum hides its structure. Once you pull out whatever every term shares, you rewrite it as a product, and products are what let you set factors equal to zero, simplify fractions, or match a pattern the test wants you to see. This is the first move to try on almost any polynomial expression: before anything fancier, ask what every term has in common.

To find the greatest common factor, work on coefficients and variables separately. For the numbers, find the largest integer that divides all of them. For each variable, take the lowest power that appears in every term. Then divide each term by that GCF and write the result in parentheses, with the GCF out front. A common factor is only valid if it divides every single term, constants included. If one term has no variable in it, a variable can never be the common factor, no matter how many other terms contain it.

The same idea scales up: sometimes the shared piece is not a single monomial but an entire binomial, like (x+2) appearing in both 5x(x+2) and 3(x+2). Treat that binomial exactly like you would treat x or 3, factoring it out front and leaving the leftover pieces behind in a second set of parentheses. This move shows up constantly right before factoring by grouping and before factoring trinomials, where pulling a common factor first often turns a messy expression into something with a recognizable quadratic or product-of-binomials shape.