- Use factor theorem to prove that (x + a) is a factor of (x" + a") for any odd positive integer n.

$$\begin{gathered} Use factor theorem to prove that (x+a)b is a factor \\ (x^n+a^n) for any odd positive integer n \\ \Rightarrow Let f\left(x\right) =x^n+a^n n is odd positive integer \\ Let take (x+a)=0 \\ x=-a \\ Let us consides f\left(x\right) =x^n+a^n \\ put x=-a \\ f(-a)={(-a)}^n+a^n \\ =-a^n+a^n \\ =0 \\ Since n is odd \\ By factor theorem \\ \mathit{H}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{ }\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{d}\mathbf{ }\mathit{t}\mathit{h}\mathit{a}\mathit{t}\mathbf{ }\mathbf{(}\mathit{x}\mathbf{+}\mathit{a}\mathbf{)}\mathbf{ }\mathit{i}\mathit{s}\mathbf{ }\mathit{a}\mathbf{ }\mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{o}\mathit{r}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{ }\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathbf{ }\mathit{n}\mathbf{ }\mathit{i}\mathit{s}\mathbf{ }\mathit{o}\mathit{d}\mathit{d}\mathbf{ } \\ \mathit{p}\mathit{o}\mathit{s}\mathit{i}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{g}\mathit{e}\mathit{r} \end{gathered}$$