Standard Form/Expansion of x^n - y^n

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In summary, there is a formula for expanding x^n - y^n which involves multiplying by a sum of terms. When dividing by yn, the expression can be simplified to (x-y) times a rearranged version of the original formula.
  • #1
Gib Z
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Just as [tex]x^a-1 = (x-1)\sum_{n=0}^{a-1} x^n[/tex], is there a similar expansion for x^n - y^n?
 
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  • #2
What happens if you divide by yn?
 
  • #3
[tex](\frac{x}{y})^n -1[/tex] which fits the previous form, but I was hoping i'd get something a bit nicer looking >.<
 
  • #4
What's wrong with it? Once you're done simplifying, it's almost the same expression. Maybe you didn't multiply the y^n back in?

I get (x - y) sum_i x^i y^(n-1-i) .
 
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  • #5
Gib Z... anyway you look at it, if you want a nice looking sum, it will only be a rearrangement of what Hurkyl proposed.
 
  • #6
Ok i see it now, my bad lol. Thanks guys
 

1. What is the standard form/expansion of x^n - y^n?

The standard form of x^n - y^n is (x - y)(x^(n-1) + x^(n-2)y + x^(n-3)y^2 + ... + xy^(n-2) + y^(n-1)).

2. How do you expand x^n - y^n?

To expand x^n - y^n, you can use the formula (x - y)(x^(n-1) + x^(n-2)y + x^(n-3)y^2 + ... + xy^(n-2) + y^(n-1)).

3. What is the difference between standard form and expansion of x^n - y^n?

The standard form is the simplified expression of x^n - y^n, while the expansion shows all the terms that are multiplied together to get to the standard form.

4. Can you give an example of the standard form/expansion of x^n - y^n?

For example, the standard form of x^3 - y^3 is (x - y)(x^2 + xy + y^2). The expansion would be x^3 - x^2y - xy^2 + y^3.

5. What is the purpose of using standard form/expansion of x^n - y^n?

The standard form and expansion of x^n - y^n can help in simplifying and solving equations, as well as identifying patterns and relationships between terms.

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