# Thread: Find the missing factor

1. ## Find the missing factor

Hi there,

I am trying to find the missing factor here:

$\displaystyle x^n - k^n = (x - k)(?)$

I have tried various things. First I tried to do it by construction, giving:

$\displaystyle x^n - k^n = (x - k)(x^{n-1} + k^{n-1} + b)$

where b is such that:

$\displaystyle xb - kb = kx^{n-1} - xk^{n-1}$

But I can't find such a b.

Then I tried to do it by long division of polynomials but because the exponent is n and not a given value, this got very messy and I couldn't get it work.

Can someone help me out?

Many thanks

2. In fact, I could say from:

$\displaystyle xb - kb = kx^{n-1} - xk^{n-1}$

[COLOR=black]That:

$\displaystyle b(x-k) = kx^{n-1} - xk^{n-1}$

$\displaystyle b = ( kx^{n-1} - xk^{n-1} )$ divided by $\displaystyle (x-k)$ (i don't know how to do division using this formatting!)

But inserting this into the identity gives a fairly trivial result and I don't think that's what the question is after...?

3. $\displaystyle x^n - k^n = (x - k)(x^{n-1} + kx^{n-2} + k^2x^{n-3} + ... + k^{n-3}x^2 + k^{n-2}x + k^{n-1})$

To verify this result, multiply it out and you will find that each term has a term that cancels with it except for the $\displaystyle x^n$ and constant terms.

4. Ahhh thankyou. I didn't think it would be something finite and clean, this makes much more sense. Many thanks and best wishes.