Polynomials - Long Multiplication

A polynomial looks like this:

example of a polynomial
this one has 3 terms

We can multiply short polynomials using Multiplying Polynomials.

But when the polynomials have 3 or more terms it is easier and more reliable to use a method like Long Multiplication for Numbers (please read that page first).

The Method

Choose one polynomial (the longest is a good choice) and then:

• multiply it by the first term of the other polynomial, writing the result down
• then multiply it by the second term of the other polynomial, writing the result underneath the matching terms from the first multiplication
• continue like that for all terms of the other polynomial
• lastly, add up the columns.

Laying the work out neatly in columns is the key, like this:

(I wrote column headings of x, x² and x³, but you don't have to)

By lining up the columns, and being careful to put the terms under the correct columns, the job becomes "automatic", and we can easily look back to see if we got it right, too.

Blank Columns

But what happens if a polynomial is missing, say, an x term or an x² term? Just leave that column blank!

Here is a more complicated example, with blank gaps:

More than One Variable

So far we have been multiplying polynomials with only one variable (x), but how do we handle polynomials with two or more variables (such as x and y)? What are the column headings?

Just ignore the columns in the question, write down the answers as they come, always checking to see if we could put an answer under a matching answer: