Before we learn how to multiply polynomials together we're going to solve a simpler problem. Here are four rectangles, one of them is 10 by 10, one of them is 7 by 10, one of them is 2 by 10 and one of them is 7 by 2.
Can we find the area of these rectangles? Of course we can 10 * 10 is 100, 7 * 10 is 70, 2 * 10 is 20, and 7 * 2 is 14.
You may have already noticed but these four smaller rectangles are combined into one big rectangle which is 17 by 12 and one way to calculate the area of the big rectangle would be to multiply 17 * 12 but another way would be to add up all of the smaller rectangles that we have. 100 + 70 + 20 + 14 makes 204 and 17 * 12 equals 204.
So what's the point? Why are we talking about this in a video on multiplying polynomials? well the way we just solve this problem is the same way we can multiply polynomials. what if we replaced those tens with “x”s.
Now the sides of our rectangle are x + 7 and x + 2 So the area would be (x + 7) * (x + 2). and we can work that the way that we just did - by looking at the parts. x times x is x squared, 7 * x is 7x, 2 * x is 2x and 7 * 2 is 14. so if we put those together we can figure out that (x + 7) * (x + 2) is x^2 + 7x + 2x + 14 which simplifies to x^2 + 9x + 14.
Now there's a way to remember this process which is called FOIL. FOIL is an acronym which stands for first, outside, inside, last.
The way that it works is it helps you remember what to multiply first to do the first terms in each parentheses which would be x * x ,then the two outside terms which would be 2 * x then the two inside terms which would be 7 * x and then the last two terms in each parentheses which would be 7 * 2. And if we put those together we get the same answer.
Now this is a really helpful way to remember everything, but we shouldn't lose sight of what's really going on. the fundamental idea is we multiply each of the terms in the first (by each of the terms in the second) and then we combine like terms. and it's important to understand this process for solving difficult problems. Let's look at one.
Let’s multiply (x^2 - 2x + 3)(x^2 + 5x - 2)
FOIL doesn't help us here - it only works for two terms times two more terms. Let's look at a picture for this one more time. We’ll make one big rectangle and break it up into all of the parts.
So we're going to have 9 terms to deal with. For the top row we get x^2 * x^2 = x^4, x^2 * -2x = -2x^3 and x^2 * 3 = 3x^2.
On the next row we get 5x * x^2 = 5x^3, 5x * -2x = -10x^2, 5x * 3= 15x.
On the last row we get -2 * x^2 = -2x^2, -2 * -2x = 4x and -2 * 3 = 6.
Now we just need to collect like terms and they're actually already organized for us to look at the diagonals.
So we get x^4 + 3x^3 -9x^2 +19x - 6,
And we’re done. And that's all there is to it when multiplying polynomials we just multiply all of the terms of 1 polynomial by all of the terms in the other and simplify.