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In this video, we're going to learn to recognize and to factor perfect square trinomials.
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A trinomial is a perfect square trinomial if it can be written in this form, where the first term of that trinomial
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can be written as the square of some expression a, and the last term can be written as the square of some expression b,
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and the middle term can be written as two times the product of a and b. In this first trinomial,
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notice we have a squared plus two a b plus b squared. This trinomial can be written in
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factored form as a plus b times a plus b. And when we have a factor that is repeated,
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it can be rewritten using exponents as the factor raised to the second power, or a plus b squared.
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In the second trinomial, we notice that the only difference here is the sign of the middle term.
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We have a squared as our first term. We have b squared as our last term.
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We have two times the product of a and b as the middle, but we have a minus sign
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in front of that second term. This perfect square trinomial can be written as the product of
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a minus b times a minus b, and using exponents, we can rewrite this as a minus b squared.
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So let's look at a couple of examples. We want to factor these perfect square trinomials. First,
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we want to recognize that that is indeed what we see is a perfect square trinomial.
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x squared plus fourteen x plus forty-nine. I can see that my first term is in fact
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something squared. It is an expression squared, or I'll write it this way as x squared. I can see
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that my last term forty-nine is something squared. It is seven squared. So in this trinomial, a is x,
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and b is seven. Two times a times b would be two times x times seven, which would be fourteen x.
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Two times x times seven. This is indeed of the form a squared plus two a b plus b squared.
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And so it factors into a plus b quantity squared. We write it as x plus seven squared.
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In our next expression, we have four w squared minus four w plus one. Let's identify first that this is a perfect square
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trinomial. Our first term, four w squared, can be written as two w squared, and our last term one can be written as
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one squared. So in this expression, a is two w, and b is one. So two times a
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times b would be four w, and I see that this is indeed four w with a minus sign in front of it.
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This form matches this expression which can be factored into a minus b
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quantity squared, and so we can write this as two w minus one squared.
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Let's continue with two more examples. We have sixteen a squared minus fifty-six a
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plus forty-nine. Notice that the first term sixteen a squared can be written as an expression
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squared. It can be written as four a squared. So sixteen a squared is a perfect square.
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Forty-nine is also a perfect square. It can be written as seven squared. And now we'll
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check the middle term here, and notice that it is indeed two times a times b,
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or two times four a times seven. There is a minus sign in front of that second term,
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and a plus sign here. And so this form matches our perfect square trinomial.
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That means that this factors into a minus b squared, or four a minus seven squared.
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In our last example, we have two x squared plus twenty x y plus fifty y squared.
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What do you notice about this trinomial first? Well, it doesn't look like it's a
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perfect square trinomial in this form because this is not a perfect square nor is fifty y
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squared. But recall that we should always begin our factoring by looking for a greatest common factor.
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What is the GCF of these three terms? Well, each of these terms is divisible by two,
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and so two is the GCF of the trinomial. And what remains after I factor the two is x squared plus
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ten x y plus twenty-five y squared. And now this trinomial is a perfect square trinomial.
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Let's check. The first term is a perfect square. It is x squared. The last term is a perfect square. It is five y
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squared. And the middle term is twice the product of a and b, or two times x times five y.
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This gives us ten x y. And the signs are positive for all terms. That means that
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this factors into a plus b squared. I still have the GCF as a factor in front of my parentheses.
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And then I have x plus five y squared. This is our completely factored form.