Multiplying Polynomials
When you multiply two polynomials, every term in the first must multiply every term in the second. For two binomials like , that is four products — and FOIL is just a checklist that keeps them in order: First, Outer, Inner, Last.
After the four products, one more step: combine like terms. The Outer and Inner products are usually a matching pair of terms, and merging them produces the middle term of your answer. Multiply four times, combine once — that is the whole method.
FOIL: four products, then combine
Take . First: . Outer: . Inner: . Last: . That gives , and combining the middle pair gives .
Signs travel with their terms. In , the second term of the first binomial is , so Inner is and Last is . Keeping the sign attached to the term is what keeps the arithmetic honest.
The box method
The box method organizes the same four products in a grid. Write one binomial's terms across the top and the other's down the side, then fill each cell with the product of its row and column. For the four cells are , , , and — identical to FOIL, just laid out visually.
The box really earns its keep on bigger products. A binomial times a trinomial makes a grid of six products, and the box guarantees you do not miss any of them.
Beyond two binomials
FOIL only works for binomial times binomial, but the underlying rule always works: multiply every term by every term, then combine like terms. For , distribute the to all three terms, distribute the to all three terms, and combine — six products in total.
A quick sanity check: the number of products before combining is the product of the term counts. Two terms times three terms should give six products. If you wrote down five, one is missing.
Worked examples
Example 1: both signs positive
Multiply .
Answer:
Example 2: a mixed-sign product
Multiply .
Answer:
Example 3: binomial times trinomial
Multiply .
Answer:
Try one yourself
Common questions
What does FOIL stand for?
First, Outside, Inside, Last — the four pairs of terms you multiply when expanding a product of two binomials. It is a memory device for "every term times every term," not a separate rule.
Can I use FOIL on a trinomial?
No — FOIL is built for exactly two terms times two terms. For anything bigger, distribute each term of the first polynomial across the whole second polynomial, or use a box with one row or column per term.
Why do the Outer and Inner terms combine?
Because they usually have the same variable part. In , Outer is and Inner is — like terms, so they merge into . If the binomials use different variables, they may not combine, and that is fine.
How do I check my expansion?
Substitute an easy number into both the original product and your answer. With , and — matching values will not certify the answer, but a mismatch always catches an error.
Want the video version?
Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.