A lot of people learn how to multiply two binomials based on what’s called the FOIL method. FOIL stands for “First,” “Outer,” “Inner,” and “Last,” and describes which pair of terms you multiply together at each step of the way. I’m going to describe a different way of dealing with the multiplication of polynomials which can show why the FOIL method works, but can also be expanded to the multiplication of any polynomials.
First, write the polynomials on top of each other with a line beneath, just like you would if you were doing normal multiplication of two numbers. If we were multiplying (a+b) and (c+d), it should look something like this:
Now you’ll do normal multiplication just like you would with numbers, except there is no carrying over, and we won’t add by columns when we’re finished with the multiplication step. You also don’t have to provide extra space once you move on to multiplying by c. You’ll also want to provide a sign for each term to tell whether it’s positive or negative so that you don’t get confused later on. For example, we would start by doing bd and ad as follows:
Next we’ll want to do the multiplication by c. Remember that we won’t add any extra space on our line. We’ll get something like this:
Now we’ll add our answer together to get our final answer: ad + bd + ac + bc. And we’re finished!
This method of multiplying helps us stay organized when we are multiplying especially large polynomials.