WEBVTT 00:21.600 --> 00:28.496 In this video, we're going to learn to recognize and to factor perfect square trinomials. 00:28.560 --> 00:37.505 A trinomial is a perfect square trinomial if it can be written in this form, where the first term of that trinomial 00:37.538 --> 00:48.916 can be written as the square of some expression a, and the last term can be written as the square of some expression b, 00:48.960 --> 00:58.526 and the middle term can be written as two times the product of a and b. In this first trinomial, 00:58.560 --> 01:05.966 notice we have a squared plus two a b plus b squared. This trinomial can be written in 01:06.000 --> 01:13.874 factored form as a plus b times a plus b. And when we have a factor that is repeated, 01:13.920 --> 01:25.386 it can be rewritten using exponents as the factor raised to the second power, or a plus b squared. 01:25.440 --> 01:32.926 In the second trinomial, we notice that the only difference here is the sign of the middle term. 01:32.960 --> 01:39.700 We have a squared as our first term. We have b squared as our last term. 01:39.760 --> 01:47.241 We have two times the product of a and b as the middle, but we have a minus sign 01:47.280 --> 01:54.014 in front of that second term. This perfect square trinomial can be written as the product of 01:54.080 --> 02:05.626 a minus b times a minus b, and using exponents, we can rewrite this as a minus b squared. 02:05.680 --> 02:12.032 So let's look at a couple of examples. We want to factor these perfect square trinomials. First, 02:12.080 --> 02:19.306 we want to recognize that that is indeed what we see is a perfect square trinomial. 02:19.360 --> 02:27.080 x squared plus fourteen x plus forty-nine. I can see that my first term is in fact 02:27.120 --> 02:37.057 something squared. It is an expression squared, or I'll write it this way as x squared. I can see 02:37.120 --> 02:54.041 that my last term forty-nine is something squared. It is seven squared. So in this trinomial, a is x, 02:54.080 --> 03:07.454 and b is seven. Two times a times b would be two times x times seven, which would be fourteen x. 03:07.520 --> 03:18.899 Two times x times seven. This is indeed of the form a squared plus two a b plus b squared. 03:18.960 --> 03:35.549 And so it factors into a plus b quantity squared. We write it as x plus seven squared. 03:35.600 --> 03:45.892 In our next expression, we have four w squared minus four w plus one. Let's identify first that this is a perfect square 03:45.926 --> 04:01.541 trinomial. Our first term, four w squared, can be written as two w squared, and our last term one can be written as 04:01.600 --> 04:15.622 one squared. So in this expression, a is two w, and b is one. So two times a 04:15.680 --> 04:27.234 times b would be four w, and I see that this is indeed four w with a minus sign in front of it. 04:27.280 --> 04:34.107 This form matches this expression which can be factored into a minus b 04:34.160 --> 04:47.320 quantity squared, and so we can write this as two w minus one squared. 04:47.360 --> 04:54.261 Let's continue with two more examples. We have sixteen a squared minus fifty-six a 04:54.320 --> 05:02.269 plus forty-nine. Notice that the first term sixteen a squared can be written as an expression 05:02.320 --> 05:12.913 squared. It can be written as four a squared. So sixteen a squared is a perfect square. 05:12.960 --> 05:21.040 Forty-nine is also a perfect square. It can be written as seven squared. And now we'll 05:21.054 --> 05:29.396 check the middle term here, and notice that it is indeed two times a times b, 05:29.440 --> 05:42.442 or two times four a times seven. There is a minus sign in front of that second term, 05:42.480 --> 05:50.817 and a plus sign here. And so this form matches our perfect square trinomial. 05:50.880 --> 06:05.632 That means that this factors into a minus b squared, or four a minus seven squared. 06:05.680 --> 06:13.006 In our last example, we have two x squared plus twenty x y plus fifty y squared. 06:13.040 --> 06:17.944 What do you notice about this trinomial first? Well, it doesn't look like it's a 06:18.000 --> 06:25.618 perfect square trinomial in this form because this is not a perfect square nor is fifty y 06:25.680 --> 06:33.793 squared. But recall that we should always begin our factoring by looking for a greatest common factor. 06:33.840 --> 06:42.202 What is the GCF of these three terms? Well, each of these terms is divisible by two, 06:42.240 --> 06:51.945 and so two is the GCF of the trinomial. And what remains after I factor the two is x squared plus 06:52.000 --> 07:03.856 ten x y plus twenty-five y squared. And now this trinomial is a perfect square trinomial. 07:03.920 --> 07:21.708 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 07:21.760 --> 07:32.752 squared. And the middle term is twice the product of a and b, or two times x times five y. 07:32.800 --> 07:45.565 This gives us ten x y. And the signs are positive for all terms. That means that 07:45.600 --> 07:54.741 this factors into a plus b squared. I still have the GCF as a factor in front of my parentheses. 07:54.800 --> 08:11.840 And then I have x plus five y squared. This is our completely factored form.