How can the expression (a^n+b^n)/(a+b) be simplified?

[eddybob123]

Hi I was wondering if there is any way to simplify this expression or if it's already in its simplest form. Thank you in advance: (a^n+b^n)/(a+b)

 

[Mandlebra]

If n is prime and a,b are elements of Z/nZ then that is (a + b)^(n-1)

 

[Curious3141]

Hi I was wondering if there is any way to simplify this expression or if it's already in its simplest form. Thank you in advance: (a^n+b^n)/(a+b)

If n is even, it cannot be simplified.

If n is odd, you can do long division without a remainder to get the following expression:

$$a^{n-1} - ba^{n-2} + b^2a^{n-3} - ... + b^{n-1}$$

which is

$$\sum_{k=0}^{n-1}a^{n-k-1}.(-b)^k$$

Of course, I wouldn't call the latter form "more simplified". The greatest exponent is lower, though.

 

[eddybob123]

@Mandlebra
n should be prime but what exactly do you mean by "Z/nZ. As you can probably see, I am an amateur mathematician.

 

[Mandlebra]

Znz is the ring of integers mod n. 0,1,2,..., n-1. Check out modular arithmetic on wiki for an idea

 

[eddybob123]

Does this work for all values of a and b (positive, negative, integral, non-integral, etc.)?

 

[micromass]

If n is prime and a,b are elements of Z/nZ then that is (a + b)^(n-1)

I don't think the OP was talking about $\mathbb{Z}/n\mathbb{Z}$. So I don't think it's good to post this kind of information since it only confuses the OP.

OP: the answer you want has been given by curiouspi.

 

[Mandlebra]

I don't think the OP was talking about $\mathbb{Z}/n\mathbb{Z}$. So I don't think it's good to post this kind of information since it only confuses the OP.

OP: the answer you want has been given by curiouspi.

Who knows. He didn't provide much info

 

[eddybob123]

What was the "Z/nZ" you were talking about in your previous post？

 

[Mandlebra]

What was the "Z/nZ" you were talking about in your previous post？

http://en.wikipedia.org/wiki/Modular_arithmetic

 

[algebrat]

What was the "Z/nZ" you were talking about in your previous post？

The numbers on a clock are Z/12Z, for example. 7+8=3 mod 12. What time is it 8 hours after 7:00?

You don't really need these numbers till you take abstract algebra (AKA advanced algebra AKA modern algebra AKA algebra, not too be confused with those of the same name for high school, or linear algebra), or number theory or maybe combinatorics.

But, if you do, then you learn that in Z/pZ, where p is prime, then you have "the freshman's dream" (a+b)^p=a^+b^p, which is based off a quick application of the binomial theorem and applying multiples of p in Z/pZ are 0 (mod p).