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## Algebra 2

### Course: Algebra 2 > Unit 1

Lesson 6: Special products of polynomials# Polynomial special products: perfect square

CCSS.Math: , ,

The perfect square pattern tells us that (a+b)²=a²+2ab+b². This can be used to expand (x+1)² as x²+2x+1, but also to expand (5x⁶+4)² as 25x¹²+40x⁶+16, or (3t²-7t⁶)² as 9t⁴-42t⁸+49t¹².

## Want to join the conversation?

- What is the F.O.I.L. method?(6 votes)
- Foil is a way of evaluating what happens when you multiply two binomials. It stands for First Outer Inner Last. What that means is if I have (x - 2)(2x + 1) we multiply the first terms, x and 2x, we multiply the outer terms,x and 1, we multiply the inner terms, -2 and 2x, and we multiply the last terms, -2 and 1. In this case the result would be 2x^2 - 3x - 2.(21 votes)

- Since2:30, I feel a bit confused. Can someone explain this a bit better please?

Thanks, I'd appreciate it.(4 votes)- This video is explaining how we can take advantage of a pattern to expand equations in the form (a + b)^2. This expands to a^2 + 2ab + b^2. If we have an equation in this form, first determine what a is and what b is. Then plug the values you get into a^2 + 2ab + b^2. Add a^2 to 2 * a * b, then add that to b^2. At2:30, sal is doing the 2 * a * b step. a = 5x^6, b = 4. If we plug the values of a and b into 2 * a * b, we get 2 * 5x^6 * 4, which is 40x^6. After that, sal does b^2, which is 4^2, which is 16. At the end, Sal ends up with 25x^12 + 40x^6 + 16. Hope this helps.(6 votes)

- wouldnt the exponent (2) multiply each terms? so for example, the problem

(5x^6 + 4)^2 would be 25x^12 + 16 since you have to multiply each terms.(2 votes)- Unfortunately a lot of people make this mistake. If you just have a number x^2 this means x*x. Now, if you have a whole term like (5x^6 + 4) and square it you are saying eerything in the parenthesis times everyting in the parenthesis. so (5x^6 + 4)*(5x^6 + 4)

It might help if you imagine everything in the parenthesis as one variable. so (5x^6 + 4) = y then (5x^6 + 4)^2 = y^2 = y*y = (5x^6 + 4) * (5x^6 + 4). From here if you multiply two things in parenthess you have to multiply each term by each term in the other. for existence, making it a little simpler, (a + b) * (c + d) = ac + ad + bc + bd. or it might help thinking that you distribute one parenthesis into the other. (a + b) * (c + d) = a(c + d) + b(c + d). I hope I'm not over explaining lol.

It might also help to try with real numbers. so try something like (2 + 3) * (5 + 7)

Let me know if this didn't help, if the explanation wasn't too late lol.(6 votes)

- so what do you do on this part of the problem 2ab?1:08(1 vote)
- If you mean how did Sal get 2ab, he just added the first ab and the second ab together, therefore getting 2ab.(7 votes)

- please where can i get a lesson on polynomial functions??(3 votes)
- Here is the video on polynomials:

https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:quadratics-multiplying-factoring/x2f8bb11595b61c86:multiply-monomial-polynomial/v/polynomials-intro(2 votes)

- At3:15in the video Sal originally writes a polynomial as (3t^2-7t^6)^2. Then he changes it to (3t^2 + -7t^6)^2. Upon first inspection it would appear to be a difference of squares not a perfect square. How would one know to add a + before -7t^6?

Thanks in advance(3 votes)- Thanks very much, that helps me understand better.(1 vote)

- so what do you do on this part of the problem 2ab?(2 votes)
- if you square a binomial (a + b) the middle term is 2*a*b no matter what they are. so in (5x^6 +4)^2 a = 5x^2 and b = 4 so 2ab = 2(5x^6)4 = 40x^6

And in 3t^2 - 7t^6 a = 3t^2 and b = -7t^6 so 2ab = 2(3t^2)(-7t^6) = -42t^8

And you would do this for all binomials that you square. You just have to identify what a and b are, but be careful you keep their signs. if one is negative the whole answer is negative, if both are positive or both are negative then the 2ab is positive.(2 votes)

- wouldnt the exponent (2) multiply each terms? so for example, the problem

(5x^6 + 4)^2 would be 25x^12 + 16 since you have to multiply each terms.(2 votes)- To square a term is to multiply it by itself.

So (5x⁶+4)²=(5x⁶+4)(5x⁶+4). You can then apply the distributive property to get 25x¹²+40x⁶+16.(2 votes)

- Question: Should you take your answer for the second question

(9t^4-42t^8+49^12) and put it into standard form?

(49t^12-42t^8+9t^4)

Or should you just leave it the way it is when you solve it?(2 votes) - is (a^2+b^2) considered a perfect square of a difference of square? or difference of square only applies to the (a^2-b^2) format?(2 votes)

## Video transcript

- [Instructor] What we're
going to do in this video is practice squaring binomials,
and this is something that we've done in the
past, but we're gonna do it with slightly more involved expressions. But let's just start with
a little bit of review. If I were to ask you,
what is a plus b squared, what would that be? Pause the video and try to figure it out. Well some of you might immediately know what a binomial like this
squared is, but I'll work it out. So this is the same thing
as a plus b times a plus b. And then we could multiply
this a times that a. So that's going to give us a squared. And then I can multiply
that a times that b, and that's going to give us ab. Then I could multiply this b times that a. I could write that as ba or ab, so I'll just write it as ab again. And then I multiply this b times that b, so plus b squared. And what I really just did is apply the distributive property twice. We go into a lot of
detail in previous videos. Some people also like to
call it the FOIL method. Either way, this should all be a review. If it's not, I encourage you to look at those introductory videos. But this is going to simplify to a squared plus we have an ab and another ab, so you add those together,
you get two ab plus b squared. Now why did I go through this review? Well now we can use this
idea that a plus b squared is equal to a squared
plus two ab plus b squared to tackle things that at least look a
little bit more involved. So if I were to ask you,
what is five x to the sixth plus four squared,
pause this video and try to figure it out. And try to keep this and this in mind. Well there is several ways
you could approach this. You could just expand this
out the way we just did, or you could recognize this pattern that we just established. That if I have an a
plus b and I square it, it's going to be this. And so what you might notice is, the role of a is being
played by five x to the sixth right over there, and the role
of b is being played by four right over there. So we could say, hey, this
is going to be equal to a squared, we have our a squared there. So what is a squared? Well five x to the sixth squared is going to be 25 x to the 12th power. And then it's going to be
plus two times a times b. So plus two times five x
to the sixth times four. Actually let me just write it out just so we don't confuse ourselves. Two times five x to the,
I'll color code it too, two times five x to the
sixth times four, times four, plus b squared. So plus four squared, so
that's going to be plus 16. And then we can simplify this. So this is going to be
equal to 25 x to the 12th. Two times five times four is 40. Two times five is 10 times four is 40. So plus 40 x to the sixth plus 16. Let's do another example. And I'll do this one
even a little bit faster, just because we're getting,
I think, pretty good at this. So let's say we're trying to determine what three t squared minus seven t to the
sixth power squared is. Pause the video and try to figure it out. All right we're going
to do it together now. So this is our a, and
our b now we should view as negative seven t to the sixth. Because this says plus b, so
you could view this as plus negative seven t to the sixth. We could even write that if we wanted. We could write this plus
negative seven t to the sixth if it helps us recognize
this whole thing as b. So this is going to be equal to a squared, which is nine t to the fourth, plus two times this times this, two times a times b. So two times three t squared
is going to be six t squared times negative seven t to the sixth. Actually let me write this out. This is getting a little bit complicated. So this is going to be plus
two times three t squared times negative seven t to the sixth power. And then last but not
least, we're going to square negative seven t to the sixth. So that's going to be negative
seven squared is positive 49, and t to the sixth
squared is t to the 12th, t to the 12th power. And so this is going to be
equal to nine t to the fourth, and let's see, two times three is six times negative seven is negative 42, and t squared times t to the
sixth, we add the exponents, we have the same base, so it's
going to be t to the eighth. And then we have plus
49 t to the 12th power. So it looks like we did
something really fancy. We have this higher degree polynomial. We were squaring this binomial that has these higher degree terms. But we're really just
applying the same idea that we learned many many many videos ago, many many lessons ago, in terms
of just squaring binomials.