What is the Expansion of X^n + Y^n When n is Even?

  • Thread starter phymatter
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In summary, the expansion of xn + yn depends on whether n is even or odd. If n is odd, it can be factored as (x+y)(x^{n-1}-x^{n-2}y+x^{n-3}y^2-x^{n-4}y^3+...-xy^{n-2}+y^{n-1}). However, if n is even, it cannot be factored over the reals and complex numbers must be used.
  • #1
phymatter
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What is the expansion of xn +yn , when is even ??/
 
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  • #2
I don't see anything that can be expanded.
 
  • #3
elibj123 said:
I don't see anything that can be expanded.

i mean that xn - yn can be written as (x-y)(xn-1 +xn-2y ...+yn-1 )
similarly what can xn +yn be written as ?
 
  • #4
Try alternating signs, and it becomes straightforward.
 
  • #5
I think you need to think about zeros
[tex]x^n+y^n=0[/tex]
[tex]x^n=-y^n[/tex]
[tex]x=y\cdot\exp(i\pi k/n)[/tex]
[tex]\therefore x^n+y^n=\prod_k (x-\exp(i\pi k/n)y)[/tex]

Occationally combining a subset of these factors together will give you a real solution.

Now you need to think when... :)
 
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  • #6
phymatter said:
What is the expansion of xn +yn , when is even ??/

phymatter said:
i mean that xn - yn can be written as (x-y)(xn-1 +xn-2y ...+yn-1 )
similarly what can xn +yn be written as ?

Then you mean what are the factors :smile:

If n is odd, you can factor it as so:

[tex]x^n+y^n=(x+y)(x^{n-1}-x^{n-2}y+x^{n-3}y^2-x^{n-4}y^3+...-xy^{n-2}+y^{n-1})[/tex]

However, if n is even, then [tex]x^n+y^n\neq 0[/tex] except for in the trivial case of x,y=0. This means you can't factor it over the reals. You'll need to use complex numbers. You could convert it into a few different ways, such as [tex]x^n-i^2y^n[/tex] and take difference of two squares, or, if you want to follow the same factorizing process as above, take [tex]x^n+(iy)^n[/tex] and take two cases, when [itexi^n[/itex] is equal to 1, and when equal to -1.
 

1. What is the formula for solving X^n + y^n?

The formula for solving X^n + y^n is (X+y)(X^(n-1) - X^(n-2)y + X^(n-3)y^2 - ... + y^(n-1)). This is known as the binomial theorem and can be used to expand and simplify this equation.

2. Can X and y be any type of number in the equation X^n + y^n?

Yes, X and y can be any type of number, including integers, fractions, decimals, and irrational numbers. The equation will work as long as both X and y are raised to the same power, n.

3. How do you solve for X and y in the equation X^n + y^n = 0?

To solve for X and y in this equation, you can use the quadratic formula. First, rearrange the equation to be in the form (X^n + y^n) = 0. Then, substitute X^n = -y^n and solve for X. This will give you two possible values for X, which you can then plug back into the original equation to solve for y.

4. What is the significance of solving X^n + y^n in mathematics?

Solving X^n + y^n is important in mathematics because it helps us understand the relationship between different types of numbers, such as integers and irrational numbers. It also allows us to manipulate and simplify complex equations, making them easier to solve and understand.

5. Are there any real-world applications for solving X^n + y^n?

Yes, there are many real-world applications for solving X^n + y^n. One example is in physics, where this equation can be used to calculate the magnitude of vector quantities. This equation is also used in engineering and computer science, as well as in financial calculations such as compound interest.

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