Main content

## Get ready for Precalculus

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

Lesson 3: Special products of polynomials# Polynomial special products: difference of squares

CCSS.Math: , ,

The difference of squares pattern tells us that (a+b)(a-b)=a²-b². This can be used to expand (x+2)(x-2) as x²-4, but also to expand (3+5x⁴)(3-5x⁴) as 9-25x⁸, or (3y²+2y⁵)(3y²-2y⁵) as 9y⁴-4y¹⁰.

## Want to join the conversation?

- Do we not need to write these in standard form? I was thinking these would be -25x^8+9 and -4y^10+9y^4(7 votes)
- It's not "necessary" but it is what is usually done most of the time. Technically either way or any order is correct(12 votes)

- i found this tricky question this is as follow

Let N be least positive integer such that whenever a non-zero digit c is written after the last digit of N, the resulting number is divisible by c. The sum of the digits of N is

i really think hard but i can't get it please anyone solve this question step wise step so i can crack it(3 votes)- Putting 𝑐 at the end of 𝑁 gives us the number 10𝑁 + 𝑐

Dividing this by 𝑐 gives us

(10𝑁 + 𝑐)∕𝑐,

which we can write as

10𝑁∕𝑐 + 1

From this follows that if 10𝑁 + 𝑐 is divisible by 𝑐,

then 10𝑁 is also divisible by 𝑐.

Now we need to minimize 𝑁 for each possible value of 𝑐, and then find the least common multiple of those 𝑁's:

𝑐 = 1 ⇒ 𝑁 = 1

𝑐 = 2 ⇒ 𝑁 = 1

𝑐 = 3 ⇒ 𝑁 = 3

𝑐 = 4 ⇒ 𝑁 = 2

𝑐 = 5 ⇒ 𝑁 = 1

𝑐 = 6 ⇒ 𝑁 = 3

𝑐 = 7 ⇒ 𝑁 = 7

𝑐 = 8 ⇒ 𝑁 = 4

𝑐 = 9 ⇒ 𝑁 = 9

LCM(1, 2, 3, 4, 7, 9) = 2 ∙ 2 ∙ 3 ∙ 3 ∙ 7 = 252

So, the least positive integer 𝑁 is 252, which has the digit sum 2 + 5 + 2 = 9(7 votes)

- So it would just be the same thing but expanded?(2 votes)
- Yes, but the important thing is to notice the pattern when you have difference of squares(4 votes)

- How does (x-2)(x+2) = x^2-2^2? When I do the equation that's what I get but I don't understand why.(1 vote)
- In general, (a+b)(a-b)=a²-b². This is because if you expand out the left-hand side, you get

a²-ab+ab-b²

The ab terms cancel, and you're left with

a²-b².(3 votes)

- isn't the last question supposed to be y to the 25th power? are you supposed to multiply them or add them? because the first question sal did multiplication and the second sal did addition.(2 votes)
- Why would you not FOIL it? It seems that when you FOIL the problem vs just distributing you get completely different answers.(1 vote)
- If you foil (x + y)(x - y), you'd get x^2 - xy + yx - y^2. xy and yx are the same thing, so you'd get x^2 - y^2. Hope this helped!(2 votes)

- Polynomial special products? What are those?(1 vote)
- are you supposed to multiply or add the exponents? in other videos and other similar equations it shows you adding them but in this video it shows you multiplying(1 vote)
- Also, these are called conjugates, which are any algebraic expression with (a+b) (a-b), which can be (5+6) (5-6) all the way up to (5x^2+7x)(5x^2 -7x), as those were the types of conjugates shown in the video.(1 vote)

## Video transcript

- [Instructor] Earlier in
our mathematical adventures, we had expanded things like x plus y times x minus y. Just as a but of review, this is going to be equal to
x times x, which is x squared; plus x times negative
y, which is negative xy; plus y times x, which is plus xy; and then minus y times y. Or you could say y times a negative y, so it's going to be minus y squared. Negative xy, positive xy, so
this is just going to simplify to x squared minus y squared. And this is all review. We covered it, and when we
thought about factoring things that are differences of squares, we thought about this when we were first learning to multiply binomials. And what we're going to do now is essentially just do the same thing, but do it with slightly more
complicated expressions. And so, another way of
expressing what we just did is we could also write something like a plus b times a minus b is
going to be equal to what? Well, it's going to be equal
to a squared minus b squared. The only difference between what I did up here and what I did over here is instead of an x, I wrote an a; and instead of a y, I wrote a b. So, given that, let's see if we can expand and then combine like terms for, if I'm multiplying these two expressions. Say I'm multiplying three
plus 5x to the fourth times three minus 5x to the fourth. Pause this video, and see
if you can work this out. Alright, well, there's
two ways to approach it. You could just approach it exactly the way that I approached it up here, but we already know that
when we have this pattern where we have something plus something times that same original something minus the other something, well that's going to be of the form of this thing squared
minus this thing squared. And remember, the only
reason why I'm applying that is I have a three right
over here and here, so the three is playing the role of the a. So, let me write that down, that is our a. And then the role of the b is being played by 5x to the fourth. So, that is our b right over there. So, this is going to be equal
to a squared minus b squared. But our a is three, so it's going to be
equal to three squared, minus, and then our b is 5x to the fourth, minus 5x to the fourth squared. Now, what does all of this simplify to? Well, this is going to be equal to, three squared is nine, and then minus 5x to the fourth squared. Let's see, 5 squared is 25. And then x to the fourth squared, well, that is just going
to be x to the fourth times x to the fourth, which
is just x to the eighth. Another way to think about
it are exponent properties. This is the same thing as 5 squared times x to the fourth squared. If I raise something to an exponent and then raise that to another exponent, I multiply the exponents. And there you have it. Let's do another example. Let's say that I were to ask you, what is 3y squared plus 2y to the fifth times 3y squared minus 2y to the fifth? Pause this video, and see
if you can work that out. Well, we're going to do it the same way. You could, of course, always just try to expand it out
the way we did originally. But we could recognize here that, hey, I have an a plus a b
times the a minus a b. So, that's going to be
equal to our a squared. So, what's 3y squared? Well, that's going to be 9y to the fourth minus our b squared. Well, what's 2y to the fifth squared? Well, 2 squared is four,
and y to the fifth squared is y to the five times
two, y to the 10th power. And there's no further
simplification that I could do here. I can't combine any like terms. And so, we are done here as well.