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Math Help - Sum or Difference between any powers.

  1. #1
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    Sum or Difference between any powers.

    I understand the difference between two squares but I don't understand this.

    <br />
a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)<br />
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  2. #2
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    Quote Originally Posted by alyosha2 View Post
    I understand the difference between two squares but I don't understand this.

    <br />
a^5 - a^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)<br />
    I would call that 0.
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    Quote Originally Posted by dwsmith View Post
    I would call that 0.

    no I mean I don't understand the process that led to the difference of those two terms being displayed in the way that it is on the right.
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    I think you mean \displaystyle a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)

    Start with \displaystyle  (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) expand it out, what do you get?

    Can you guess the following?

    \displaystyle a^3 - b^3 = \dots

    \displaystyle a^4 - b^4 = \dots
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  5. #5
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    Quote Originally Posted by alyosha2 View Post
    no I mean I don't understand the process that led to the difference of those two terms being displayed in the way that it is on the right.
    Technically the RHS is 0 since a^5-a^5=0
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  6. #6
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    Quote Originally Posted by dwsmith View Post
    Technically the RHS is 0 since a^5-a^5=0
    sorry, my mistake. I meant the second a to be a b.
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    When you say expand do you mean factor out?
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  8. #8
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    Quote Originally Posted by pickslides View Post
    I think you mean \displaystyle a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)

    Start with \displaystyle  (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) expand it out, what do you get?

    Can you guess the following?

    \displaystyle a^3 - b^3 = \dots

    \displaystyle a^4 - b^4 = \dots
    when you say expand out do you mean factor out?
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  9. #9
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    Spoiler:
    \displaystyle a^n - b^n = (a - b)(a^{n-1}b^0 + a^{n-2}b^1 + \dots +a^0 b^{n-1})
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  10. #10
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    Expand \displaystyle (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4) and simplify. Does it equal \displaystyle a^5 - b^5?


    Can you guess the factorisations of \displaystyle a^3 - b^3 and \displaystyle a^4 - b^4? Can you see a pattern?
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  11. #11
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    Quote Originally Posted by alyosha2 View Post
    when you say expand out do you mean factor out?
    I'm asking you to multiply the brackets out. There is a pattern here, can you find it?
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  12. #12
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    Here is what I understand.

    <br />
a^5 - b^5 = (a + b)(a^4 - b^4) = (a + b)(a + b)(a^3 - a^3) = (a + b)(a + b)(a + b)(a^2 - b^2) = (a + b)(a + b)(a + b)(a + b)(a - b)<br />

    but I don't understand how we get to the right of what I first posted.
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  13. #13
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    a^5-b^5 \neq (a+b)(a^4-b^4)

    Prove it! and I are suggesting you start with the RHS to show the LHS, maybe this might clear some things up.
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    Quote Originally Posted by pickslides View Post
    I'm asking you to multiply the brackets out. There is a pattern here, can you find it?
    I can see the pattern but I don't understand how we get to this way of expressing the difference between the first two powers.
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  15. #15
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    I get to

    <br />
a^5b + a^4b^2 + a^3b^3 + a^2b^4 + ab^5<br />
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