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### Course: Get ready for Precalculus>Unit 2

Lesson 3: Special products of polynomials

# Polynomial special products: perfect square

Squaring binomials is a breeze when you recognize patterns! The perfect square pattern tells us that (a+b)²=a²+2ab+b². The video shows how to square more complex binomials. It's all about applying what we know about simple binomials to these trickier ones.

## Want to join the conversation?

• What is the F.O.I.L. method?
• 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.
• Since , I feel a bit confused. Can someone explain this a bit better please?
Thanks, I'd appreciate it.
• 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. At , 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.
• 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.
• 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.
• At in 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 very much, that helps me understand better.
• so what do you do on this part of the problem 2ab?

• If you mean how did Sal get 2ab, he just added the first ab and the second ab together, therefore getting 2ab.
• When calculating the final answer, does the answer need to be in highest degree to lowest degree?
• Not necessarily. It is common practice, and it makes things simpler in a lot of cases, especially having the highest degree as the first term. Some teachers may require it, but in general it isn't something that HAS to be done.
• please where can i get a lesson on polynomial functions??
• I'm a tiny bit confused: If the method is (a+b)^2 is equal to a^2+2ab+b^2, then wouldn't the exponents turn out differently?

For example: (5x^6+4)^2. Without using Sal's method, I'd do 5x^6(5x^6+4) + 4(5x^6+4). This turns out to be 25x^36+20x^6+20x^6+16. This simplifies to 25x^36+40x^6+16. Easy, right?

But then I go to try with Sal's method of a^2+2ab+b^2, and saw that his result was 25x^12+40x+16.

The difference being that the exponent for mine was 36, his was 12.

Why is this, if the two methods are supposed to give the same result? What did I do wrong?

Thanks.
• When you multiply numbers that are raised to a power, the exponents are added, not multiplied. So, 5x^6 * 5x^6 = (5 * 5) * (x^6 * x^6) = 25 * x^(6 + 6) = 25x^12.
• so what do you do on this part of the problem 2ab?
• 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.