Factoring Polynomials

Reverse distribution: pull out common factors, then group.

The idea

Factoring rewrites a sum as a product by running the distributive property backwards: $ab + ac = a(b + c).$ As areas: a rectangle of height $a$ and width $b + c$ is made of two cells of areas $ab$ and $ac$, as the figure shows. The product form reveals what the sum conceals — above all, a product is zero exactly when one of its factors is, which is what makes factored equations solvable.

The first step in any factoring problem is to pull out the greatest common factor: the largest number dividing every coefficient, together with the highest power of each variable that appears in every term. Take all of it at once; leaving part behind only means factoring again.

An expression whose terms share no common factor may still factor. With four terms, try grouping: split them into two pairs and factor each pair separately. If both pairs leave behind the same binomial, that binomial is a common factor of the two groups, and pulling it out completes the factorization. If not, try a different pairing before concluding the expression does not factor.

Ways to work on it

• Walkthrough. Greatest common factors and factoring by grouping.
• Practice. Factor a binomial by pulling out its greatest common factor.
• Hardest. Factor a four-term expression by grouping.

Not sure where to start? Take the ten-question placement test.