from my text book it writes: [see pic]

i don't know how could it prove (2)and(3). why doesn't it show to proving?? i know factor thm but i just don't understand how are they derived from factor theorem. many thanks!

- May 17th 2008, 09:47 AMafeasfaerw23231233polynomial : x^n - a^n divided by x + a , x^n + a^n divided by x+a
from my text book it writes: [see pic]

i don't know how could it prove (2)and(3). why doesn't it show to proving?? i know factor thm but i just don't understand how are they derived from factor theorem. many thanks! - May 17th 2008, 11:15 AMflyingsquirrel
Hi

(2) : As $\displaystyle n$ is a positive even integer, it can be written as $\displaystyle n=2k$ for $\displaystyle k\in\mathbb{N}$.

$\displaystyle f(x)=x^n-a^n=x^{2k}-a^{2k}=(x^2)^k-a^{2k}$

Hence, $\displaystyle f(-a)=((-a)^2)^k-a^{2k}=(a^2)^k-a^{2k}=a^{2k}-a^{2k}=0$. As $\displaystyle -a=\frac{-a}{1}$, $\displaystyle x^n-a^n$ can be factored by $\displaystyle 1\cdot x-(-a)=x+a$ thanks to the factor theorem.

Does it help ?