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## Integrated math 3

### Course: Integrated math 3>Unit 1

Lesson 6: Special products of polynomials

# Polynomial special products: difference of squares

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
• It's not "necessary" but it is what is usually done most of the time. Technically either way or any order is correct
• 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
• 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
• So it would just be the same thing but expanded?
• Yes, but the important thing is to notice the pattern when you have difference of squares
• 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.
• 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.
• 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!
• 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)
• What is x^2-14x+49
(1 vote)
• (x-7)^2
(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)