Hey everyone, it's Justin again.
In the last lesson, you learned how to multiply polynomials using a grid,
but it's time to ditch that grid and learn how to multiply polynomials algebraically! In this video,
we're going to learn about a new method called the F.O.I.L. method. At the end of this lesson,
you will be able to multiply binomials and other polynomials algebraically. To get there first,
you'll learn what the acronym F.O.I.L. stands for and what it means, and then we'll use the
F.O.I.L. method to work out two examples. Because F.O.I.L. only applies to binomials,
we'll then expand the pattern of F.O.I.L. so it can be used with larger polynomials.
Let's do this example of a binomial multiplication using the grid method,
but this time let's keep track of which ones we're multiplying. The first space has us
multiply the first term in each binomial. The next space on that row multiplies the x from
the first binomial with the 5 from the second, so we multiply the two terms on the outside.
Now we move to the next row and multiply the -3 and x which are the two innermost terms,
and on our final space we multiply the last term inside each set of parentheses.
Any time you multiply two binomials, you'll end up performing these four multiplications.
First, Outer, Inner, and Last. The order isn't necessarily important as long as all four
are done, although as we multiply bigger and bigger polynomials, you'll come to appreciate
the structure of doing it in the same order every time. F.O.I.L.: First, Outer, Inner, Last.
Let's try using the F.O.I.L. method without the grid. F is for first. We multiply the
first term of each binomial, 2x times 3x. O is for outer. Multiply the two outermost terms.
I is for inner. Multiply the two innermost terms. And finally, L is for last. Multiply the last
term of each binomial. The last thing to do if necessary, and it is necessary for this example,
is combining like terms. So 2x + 3 times 3x - 1 is 6x² + 7x - 3. One more example.
Pause the video now and try this one yourself.
Let's see how you did. First,
Outer, Inner, Last. This time there are no like terms so our answer stays like this.
I know I said that that was the last example, but let's go a little deeper. You might have noticed
already but we don't really need the F.O.I.L. acronym, there's a clear pattern in play here.
We start with the first term and multiply it by each term of the second binomial in order.
Once you get to the end, move to the next term in the first binomial and do the same thing,
multiplying down the list of terms for that second binomial. And we do this for every term in the
first polynomial! Recognizing this distribution pattern is super important because it can be
used when multiplying larger polynomials. And this trinomial times binomial multiplication,
we would need to multiply six times, which is two more times than F.O.I.L. has letters for. But by
using the distribution pattern we just mentioned, we can multiply these, or any polynomials!
We start with the first term and then multiply it by each term of the second polynomial Then move
on to the next term and multiply it by the second polynomial again. And then one more time for the
final term. And of course, we combine like terms at the end. It looks like a lot of steps when
you first learn it, but it's actually a really simple process that you will master in no time!
Just to recap. If it's just a simple binomial times binomial you can use the F.O.I.L. method to
remember which terms to multiply: First, Outer, Inner, Last, but don't become too reliant on
it because to multiply any larger polynomials you'll have to know the distribution pattern.
First term times the second polynomial. Second term times the second polynomial. Third term
times the second polynomial, until you've gone through every term in that first polynomial.
And remember, that pattern also works for binomials so it's actually more like this.
Now that you know a few methods for multiplying polynomials, in the next few videos we're
going to be learning about some of the special results that can sometimes occur. See you then!