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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.

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  • blobby green style avatar for user mikmic23
    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)
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  • stelly blue style avatar for user aniketprasad123
    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)
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    • cacteye blue style avatar for user Jerry Nilsson
      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)
  • leafers ultimate style avatar for user Violet Evergarden
    So it would just be the same thing but expanded?
    (2 votes)
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  • male robot donald style avatar for user aeppa
    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)
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  • blobby green style avatar for user Dylan Wang
    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)
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  • duskpin tree style avatar for user Midnight the wolf
    Polynomial special products? What are those?
    (1 vote)
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  • aqualine tree style avatar for user Vicky
    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)
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  • male robot johnny style avatar for user HarryM
    Why would you not FOIL it? It seems that when you FOIL the problem vs just distributing you get completely different answers.
    (1 vote)
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  • hopper cool style avatar for user Rehmatullah Masood
    What is x^2-14x+49
    (1 vote)
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  • old spice man green style avatar for user Jarod
    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)
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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.