# Multiplying polynomials

Multiplying polynomials is easy enough, but it can get a bit messy. Especially when dealing with a few variables. But if you know what you are doing, you will manage quite nicely. So let us get down to business.

There is a simple logic behind **multiplying polynomials** – just multiply every term in the first polynomial with every term of the second polynomial. After that, just tidy up the remaining terms by performing the necessary mathematical operations, such as addition and subtraction. Apart from that, all other rules for multiplication and the order of operations still apply and they should be observed.

Example 1.

So, if you have a polynomial like this:

(4m + 3) * (3m – 2m)

…the first thing you should do is to multiply the terms from the first polynomial with each term in the second one. The process looks like this:

4m*3m + 4m *(-2m) + 3*3m + 3*(-2m)

12m^{2} – 8m^{2} + 9m – 6m

After a bit of tidying up, it should look like this:

4m^{2} + 3m

And that is it. That is the process of *multiplying polynomials*. It is easy, right? Now we are going to solve a bit more complicated example to show you how to deal with the clutter that appears in these cases.

Example 2.

Let us assume that we have to simplify the product of these polynomials.

(-a + 3b) * (-a^{2} + ab + 3b^{2})

Again, we have to start by multiplying each term from the first polynomial with the terms in the second one.

(-a)*(-a^{2}) + (-a)*ab + (-a)*3b^{2} + 3b^{2}*(-a^{2}) + 3b*ab + 3b*3b^{2}

a^{3 }- a^{2}b – 3ab^{2} – 3a^{2}b + 3ab^{2} + 9b^{3}

Now it is time to perform the addition and subtraction to get this mathematical expression in order. Keep in mind that these operations can only be performed with terms whose variables are exactly the same. We will rewrite this expression in a way that these variables are next to each other. So, we get something like this:

a^{3} – a^{2}b – 3a^{2}b – 3ab^{2} + 3ab^{2} + 9b^{3}

We can leave out the two terms that have the same value, but opposite signs since their sum is 0. That means the result of our simplification is:

a^{3} – 4a^{2}b + 9b^{3}

As you can see, even the most complicated examples are not that difficult to solve. However, a considerable amount of concentration is required because mistakes can happen pretty easily. When you deal with __multiplying polynomials__, be sure to check your calculations before going further with an assignment. It is worth the extra effort.

So, this is all there is to multiplying polynomials. They can get more complicated by adding more variables or extra polynomials, but if you follow these basic rules and focus on your calculations, you can solve them all. If you wish to practice multiplying polynomials, feel free to use the worksheets below.