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[SOUND] Let's look at multiplying binomials.

For example if we wanted to multiply 2x + 5y and 3x - y, we could take this entire

binomial here. And distribute it to each of these two

terms, which would give us 2x + 5y * 3x and then + 2x + 5y * -y.

And again we can distribute, this 3x to these two terms.

As well as the -y, to these two terms which gives us 2x * 3x + 5y * 3x + 2x *

-y and + 5y * -y. Which is equal to 6x^2 + 15yx - 2xy -

5y^2. And now 15yx and -2xy are like terms

because yx and xy are equal by commutativity.

So, we can combine them, which gives us our answer of 6x^2 + 13xy - 5y^2.

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Now this type of multiplication comes up often, and there is an acronym used to

describe it. [SOUND] And the acronym is FOIL.

The F in FOIL stands for first, which means we multiply the first terms in each

binomial so a * c. The O stands for outer which means we

multiple the outer terms, in the two binomials,

so a * d. The I stands for inner.

Which means we multiply the inner terms of the two binomials,

so b * c. And finally the L here stands for last.

Which means we multiply the last terms in the two binomials,

so b * d. And the answer to this multiplication is

the sum of all of these. F + O + I + L.

So let's apply this method here, to see that we get the same answer that we just

found, okay?

So, we still have this product, 2x + 5y * 3x - y.

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So, what is the F here? This is a product of the first terms in the two binomials.

So 2x * 3x = 6x^2. And what about O?

O is a product of the outer terms in the two binomials.

So 2x * -y = -2xy. And I, is the product, of the inner

terms, in the two binomials. So 5y and 3x = 15yx.

And L, is the product of the last terms, in the two binomials.

So 5y * -y = -5y^2 and therefore this product is the sum of all these, so it's

6x^2 - 2xy + 15yx - 5y^2. And again, -2xy and 15yx are like terms,

so we can combine them. Which gives us, the same answer of 6x^2 +

13xy - 5y^2. So FOIL is a quick way to do distributive

multiplication of two binomial. Let's see another example.

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Let's multiply these two binomials here. Well, we can apply the FOIL method we

just saw. This is equal to the product of the first

terms in the two binomials so 2x * 2x + the product of the outer in the two

binomials. So 2x * 1 and then plus the product of

the inner terms in the two binomials. So -1 * 2x and then plus the product of

the last terms in the two binomials, so -1 * 1 which gives us 4x^2 + 2x - 2x - 1.

And notice the outer and the inner terms will cancel.

Which leaves us with our answer of 4x^2 - 1.

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Now it should be pointed out, that these binomials are special and that we're

multiplying together the difference and sum of the same two terms.

We have 2x and 2x and 1, and 1. And there's a special formula in this

type of case. And the formula, is that A - B * A + B =

A^2 - b^2 because the outer and inner terms will always cancel, which is what

we just saw. That is, in our case, our A = 2x and our

B = 1. So our answer here is A^2 - b^2 or 2x^2 -

1^2 which is this 4x^2 - 1. Let's do another example.

Now be careful here with this power of 2, we cannot apply that to each of these two

terms. What this means is 3y - 5 * 3y - 5.

So, again we can FOIL. This is equal to the product of the first

terms in the two binomials, so 3y * 3y and then plus the product of the outer

terms in the two binomials, so 3y * -5 and then plus the product of the inner

terms in the two binomials. So -5 * 3y and then plus the product of

the last terms in the two binomials. So -5 * -5 which is equal to 9y^2 - 15y -

15y + 25. And combining the outer and inner terms

gives us our answer of 9y^2-30y+25. Now again, this is the common type of

multiplication here where we're multiplying a binomial by itself and

there's a special formula again in this type of case.

And the formula is that (A - B)^2 = A^2 - 2AB + B^2.

That is the outer and inner terms are the same, so there'll be two of them, which

we just saw with A = 3y and b = 5. So, our answer here is A^2 or 3y^2 - 2 *

A * B + B^2 or 9y^2 - 30y + 25. And this how we multiplied two binomials.

Thank you, and we'll see you next time. [SOUND]