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

Lesson 3: Special products of polynomials

# Polynomial special products: difference of squares

Dive into the exciting world of special products of polynomials, focusing on the difference of squares. We explore how to expand and simplify algebraic expressions. We also tackle more complex expressions, applying the same principles to make math magic happen!

## 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
• Why would you not FOIL it? It seems that when you FOIL the problem vs just distributing you get completely different answers.
• 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!
• 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
• Ok so this might be a dumb question lol but... is the answer (3)^2-(5x^4)^2 or is the answer 9 - 25x^8 I understand both are equal but what I'm asking is which answer should I choose if I were actually doing this problem?? Sal probably specified that, but I didn't hear it and I just wanna be sure...
• The answer is the second one. 3^2-(5x^4)^2 is the less simplified version, he just uses that to show all his steps and make it easier to understand. This first expression simplifies to 9-25x^8, as shown in the video, which is the correct answer. Always try to get as simple as possible.
• 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.
• 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.
• What software and writing tool did you use to make these videos?
(1 vote)
• what does sal mean by special products