WEBVTT
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In this section, we're going to factor binomials. Specifically, we're going to look at an expression
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called the difference of two squares. A binomial is a difference of two squares
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if the first term can be written as a squared, or a perfect square, and the second term
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can be written as b squared, and there is a subtraction between these two terms.
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a squared minus b squared is a difference of two squares, and it can be factored into a plus b
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times a minus b. We learned in a previous section that if I use the distributive property
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to multiply a plus b times a minus b that I would end up with this difference of two squares.
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x squared minus nine is an example of a binomial that is a difference of two squares because the
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first term is a perfect square, the second term is a perfect square, and there is a difference or
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a subtraction between these two terms. Thus, it factors into x plus three times x minus three.
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Let's take a look at some examples. We want to factor the binomial completely for each of
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these expressions. b squared minus twenty-five is our first example. We notice that the first term
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is a perfect square, b squared, and twenty-five is a perfect square. It can be written as five
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squared. And we are subtracting them. Thus, this can be written as b plus five times b minus five.
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b squared minus five squared gives me b plus five times b minus five.
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In our second example, we have eighty-one x squared minus sixteen. We notice first that our
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first term can be written as a perfect square. Eighty-one x squared can be written as nine x
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squared. And then sixteen can be written as four squared, and there is a subtraction between them.
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So now we recognize this as the difference of two squares that can be written in factored form as
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a, being nine x, plus b, which is four, times nine x minus four. a plus b times a minus b.
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In our next example, we have four x squared minus forty-nine y squared. The first term can be written or is a perfect square,
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and it can be written as two x squared. This last term is a perfect square and can be written as seven y
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squared. And there's a subtraction between these two terms. So this is the difference
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of two squares. Thus, it factors into two x plus seven y times two x minus seven y. And
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remember the order of these factors does not matter because multiplication is commutative.
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Let's look at our last example. Fifty x squared minus two. Doesn't look like this
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is a difference of two squares in this form. But remember we must first look for a common factor,
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if possible, and I notice that fifty x squared and two have a common factor of two. So we'll
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begin by factoring out the GCF of two which leaves us with twenty-five x squared minus one.
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And this binomial is the difference of two squares because twenty-five x squared is the
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same as five x squared and one is the same as one squared. And so this binomial factors into five
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x plus one times five x minus one. And don't forget we still have the common factor two that belongs in this
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factored form as well. And so this binomial factors into two times five x plus one times five x minus one.
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Let's continue by factoring these binomials completely. Our next example is sixty-four n
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squared minus thirty-six. You may recognize this as a difference of two squares,
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but first remember that we always want to begin our factoring by factoring out a common factor,
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or a GCF, if possible first. That will make our factoring more efficient, so looking at these two
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terms sixty-four n squared and thirty-six, the GCF of these two terms is four. So if I factor
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a four out of each term, that will leave me with sixteen n squared minus nine. Sixteen n squared
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is a perfect square, and nine is a perfect square, and there is a subtraction between these two terms,
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meaning that this is the difference of two squares and can be factored into a plus b
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times a minus b. So our final factored form will include the four as our GCF
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and then a plus b where a is four n and b is three. So four n plus three,
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and a minus b which would be four n minus three. So our completely factored form of this binomial
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is four times four n plus three times four n minus three. Our next example is sixty-four n squared plus thirty-six. Let's begin
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also by looking for a common factor. You may notice that this example looks very
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similar to the one that we just did. The only difference here of these two examples is the
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sign. So we know that we can factor a GCF of four from these terms. This gives us four
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times sixteen n squared plus nine. Now looking at the binomial that remains here,
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we have a perfect square and a perfect square, but this is not the difference of two squares
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because there is a sum or a plus between these two terms. The sum of two squares cannot be factored.
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So this is our completely factored form of this binomial. Our next example is y to the fourth minus eighty-one.
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There is no common factor between these two terms, but I can recognize that each of these terms is a perfect square.
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y to the fourth can be written as y squared squared, and eighty-one can be written as nine squared,
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and there is a subtraction between these two terms making this a difference of two squares.
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So this difference of two squares factors into y squared plus nine times y squared minus nine. Now let's look a little closer
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at these two binomials that remain. y squared plus nine is the sum of two squares. We just
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talked about the fact that we cannot factor the sum of two squares in the real number system.
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y squared minus nine can be factored it is a difference of two squares and can be factored
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as y plus three times y minus three. So our final factored form of y to the fourth minus
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eighty-one will be y squared plus nine times y plus three times y minus three.
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And our last example is negative t cubed plus t. I have two terms. I want to begin by looking
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for the GCF of these two terms, and when my expression begins with a negative recall that
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it's often convenient to factor out a negative. So we will factor out a common factor of negative t.
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This will leave us with a positive t squared and a minus one. You can check that GCF factoring by redistributing
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through your grouping symbols. And now I recognize that the binomial that remains, t squared minus
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one, is the difference of two squares, and it can be factored into t plus one times t minus one.
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And let's not forget our GCF of negative t. So our final factored form is negative t times t plus one times t minus one.