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Special Products of Binomials

Special Products of Binomials

Author: Colleen Atakpu
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This lesson covers special products of binomials.

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Today, we're going to talk about special products of binomials. Special products of binomials are useful when you are factoring or FOILing expressions, so that you can more quickly figure out what the factored or FOILed form would be. They're shortcuts for factoring or FOILing.

So we're going to look at a couple of these special products of binomials, and then we'll do some examples.

So using FOIL, I'm going to start by multiplying my first two terms, 3x times negative 4x. And that's going to give me negative 12x squared. Then, I'll multiply my outside terms, 3x and 2, which will give me positive 6x. Then, I'll multiply my inside terms. Negative 1 times negative 4x will give me a positive 4x.

And finally, I'll multiply my last two terms. A negative 1 times 2 will give me negative 2. I can then combine my two like terms. 6x plus 4x will give me 10x. It's positive. And then, I'll bring down my other two terms. So the multiplication of these two binomials gives me negative 12x squared plus 10x minus 2.

So let's look at some examples of perfect square trinomials. So here, we have two examples of what we call perfect square trinomials. I'm going to expand them to show you why that's what we call them.

So I know that x plus 5 squared is the same as x plus 5 times x plus 5. That's what squaring something means, multiplying it by itself.

And written in this form, I can use FOIL to multiply the two binomials. x times x is going to give me x squared. x times 5 will give me 5x. 5 times x will give me another 5x, and 5 times 5 will give me 25. Adding together my two middle terms, I have x squared plus 10x plus 25.

So in expanded form, I can see that my expression has three terms. Because it has three terms, we call it a trinomial. And we call this a perfect square trinomial because we can write this either as x plus 5 squared or x plus 5 times x plus 5. So we call these, either in factored form or expanded form, perfect square trinomials.

Another name that we have for perfect square trinomials with a plus sign in between the two terms is the square of a binomial sum. So this is an example of a square of a binomial sum.

We can generalize this by looking at the expression x plus a, where we have any value for a, squared. Following our same process, we can come up with a general form and expanded form.

I'm going to use FOIL to multiply x squared, a times x, another a times x. And finally, a times a will give me a squared. Simplifying by combining my two middle terms, because they're like terms, this will give me x squared plus 2ax plus a squared.

So I can generalize my expressions for the square of a binomial sum using either this general form in factored form or this general form in expanded form.

Let's look at an example of a perfect square trinomial with a minus sign in between the two terms. I'm going to rewrite this in factored form as x minus 4 times x minus 4.

I, again, will use FOIL to multiply my two binomials. x times x will give me x squared, x times negative 4 will give me negative 4x, negative 4 times x will give me another negative 4x, and negative 4 times negative 4 will give me positive 16. I can combine my two middle terms, which will give me x squared minus 8x plus 16.

So again, we see that our expanded form has three terms, which makes it a trinomial. And again, because we can write it either as x minus 4 squared or x minus 4 times x minus 4, this makes it a perfect square trinomial.

I can write this in general form. This is also, besides a perfect square trinomial, called the square of binomial differences. So I can come up with a general form for the square of binomial differences.

So if I have x minus any value for a squared, I can again write this as x minus a times x minus a. Using FOIL to multiply, I'll have x squared minus ax minus another ax. And this will give me a positive a squared.

Combining my two middle terms, I have x squared minus 2ax plus a squared. And this gives me the general form for the expanded form of the square of binomial difference. And this is the factored form.

So now, let's look at another special product of binomials, which is called the difference of squares. An example would be x plus 3 times x minus 3. I'm going to FOIL, and then I'll show you why we call it the difference of squares.

So using FOIL I'll start by multiplying x times x, which will give me x squared. Then, x times negative 3 will give me negative 3x. 3 times x will give me positive 3x, and 3 times negative 3 will give me a negative 9.

I can combine my two middle terms, negative 3x and positive 3x, but I see that those two are going to cancel each other out, because negative 3 plus 3 will equal 0. So I'm left with x squared minus 9 for my expanded form of this expression.

And so again, we call this the difference of squares because both x is squared and the constant term from our original expression, negative 3 or positive 3, is also squared to give us 9. And we call it the difference because we are subtracting those two terms.

So we can come up with a general form for using the difference of squares, either written in factored form or expanded form.

So the general form would look like x plus some value a multiplied by x minus the same value of a. Using FOIL to expand and multiply, I'll have x squared minus a times x plus a times x minus a times a, or a squared.

Again, I see that my middle two terms are going to cancel. And this will always be the case, because the coefficients will be the same, just with opposite signs. So they will we cancel each other out. So this will give me x squared minus a squared. So this is the general form of the expanded form of our difference of squares.

It's important to notice that we will also have a difference of squares if we have x minus a times x plus a. This written in expanded form will also bring us back to this expression.

And that's because with multiplication, it's commutative. The order doesn't matter. So we can have x minus a times x plus a, or we can switch the order. Either will give us this expression in expanded form.

So knowing and recognizing the special products of binomials will help you FOIL and factor much faster. For example, if I have the expression x plus 3 squared, I recognize this as a perfect square trinomial.

And so using the general form for a perfect square trinomial, instead of going through the steps of multiplying, I can simply write this in expanded form to be equal to x squared plus 2 times my constant value, or 6, and then my x. And then, plus my constant value squared, so 9.

I also recognize this to be the difference of squares. So using that general form, I can rewrite this to be x squared minus 49.

So let's go over our key points from today. FOIL is the acronym used to remember the way to multiply two binomials. It stands for First, Outside, Inside, Last.

Recognizing special products of binomials can help make factoring and FOIL easier. And a perfect square trinomial is a polynomial with three terms that can be simplified as a binomial squared.

So I hope that these key points and examples helped you understand a little bit more about special products of binomials. Keep using your notes and keep on practicing, and soon, you'll be a pro. Thanks for watching.

Notes on "Special Products of Binomials"

Key Formulas

left parenthesis x plus a right parenthesis squared equals x squared plus 2 a x plus a squared

left parenthesis x minus a right parenthesis squared equals x squared minus 2 a x plus a squared

left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis equals x squared minus a squared

Key Terms

  • Perfect square trinomial: 
  • A polynomial with three terms, which can be simplified as a binomial squared, left parenthesis x plus a right parenthesis squared.
  • Difference of squares: 
  • Two squared terms separated by subtraction,x squared minus a squared which can be expressed by left parenthesis x plus a right parenthesis times left parenthesis x minus a right parenthesis.