Multiplication of Polynomials
If you understand the distributive property, then learning to multiply polynomials is going to be a piece of cake. Before we get into multiplying full-blown polynomials, let’s start with just multiplying two terms. Suppose we have the following:
Because everything is being multiplied together here, we can rearrange each factor in whatever order we want. The idea is to rearrange each factor in an order that will allow us to simplify everything. We’re going to use extra parentheses in our examples to make it easier to see the grouping, and doing so will help you to organize your work until you get used to this. So for our example, when multiplying the two terms together, we should have something like:
And that’s all there is to it. You’ll notice here that understanding the rules for exponents is going to be very helpful. So let’s look at a slightly more complicated form of multiplication, multiplying a term by a small polynomial.
The first step is going to be to use the distributive property. Remember that the distributive property says that a(b+c) = ab + ac. Here we go:
Now we know that multiplication comes before addition in the order of operations, so now we multiply the terms together as called for and see what we get:
From this point, we cannot add the two terms together because there are no like terms, and we have simplified the original polynomial as much as possible.
Going one step forward in this process, we can multiply two full-blown polynomials together:
The first step is going to be to use the distributive property. It’s a little more difficult to see here, so we’re going to break it up in a few extra steps to clearly show what is what when the distribution is being done. Let’s start with the distributive property definition:
With this clarified, let’s now fill in the blanks in the distribution and simplify our equation.
If you notice, now it’s like we have two different problems like the previous example being added together. We already know how to do these, so let’s hop into the distributive property and go from there.
All that’s left is multiplying individual terms together, so we’re home free.
And there are no like terms, so we don’t have anything else to simplify, and we’re finished.