Multiplying Polynomials

A polynomial looks like this:

example of a polynomial
this one has 3 terms

To multiply two polynomials:

• multiply each term in one polynomial by each term in the other polynomial
• add those answers together, and simplify if needed

Let us look at the simplest cases first.

1 term × 1 term   (monomial times monomial)

To multiply one term by another term, first multiply the constants, then multiply each variable together and combine the result, like this (press play):

(Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")

For more about multiplying terms, read Multiply and Divide Variables with Exponents

1 term × 2 terms   (monomial times binomial)

Multiply the single term by each of the two terms, like this:

2 term × 1 terms   (binomial times monomial)

Multiply each of the two terms by the single term, like this:

(I did that one a bit faster by multiplying in my head before writing it down)

2 terms × 2 terms (binomial times binomial)

That is 4 different multiplications ... Why?

It is the same when we multiply binomials!

Instead of Alice and Betty, let's just use a and b, and Charles and David can be c and d:

We can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms.

But there is a handy way to help us remember to multiply each term called "FOIL".

It stands for "Firsts, Outers, Inners, Lasts":

Firsts: ac Outers: ad Inners: bc Lasts: bd

So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc.

Let us try this on a more complicated example:

2 terms × 3 terms (binomial times trinomial)

"FOIL" won't work here, because there are more terms now. But just remember:

Multiply each term in the first polynomial by each term in the second polynomial

Like Terms

And always remember to add Like Terms:

Long Multiplication

You may also like to read about Polynomial Long Multiplication