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Polynomials - Long Multiplication

A polynomial looks like this:

polynomial example
example of a polynomial
this one has 3 terms

You can multiply small polynomials using the method shown on Multiplying Polynomials

But when the polynomials have 3 or more terms it is often easier (and you are less likely to make a mistake) when you use a method similar to Long Multiplication for Numbers.

The Method

Lay your work out neatly in columns (I wrote column headings of x, x2 and x3, but you don't have to) and multiply like this:

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

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 you handle polynomials with two or more variables (such as x and y)? What are the column headings?

The only method I know is to ignore the columns in the question, write down the answers as they come, always checking to see if you could put your answer under a matching answer: