Question

Use the Factor Theorem to prove that $x+c$ is a factor of

role="math" localid="1646067091231" $x^n+c^n$ if $n\ge 1$ is an odd integer

Short answer

If $f\left(x\right)=x^n+c^n$and n is a positive odd integer then $f(-c)=0$so$x+c$is a factor of $x^n+c^n$whenn is a positive odd integer

Step-by-step solution

Step 1. given data 

The given expression is

$x^n+c^n$

Factor is$x+c$

Step 2. proof 

Consider function$f\left(x\right)=x^n+c^n$

According to Factor Theorem, If $x+c$is a factor of function then$f(-c)=0$

substitute c for x in the function

$$\begin{gathered} f\left(x\right)=x^n+c^n \\ f(-c)={(-c)}^n+c^n \end{gathered}$$

n is a positive odd integer

localid="1646067391807" $$\begin{gathered} f(-c)=-c^n+c^n \\ f(-c)=0 \end{gathered}$$

the function value is zero for $x=-c$

so $x+c$is a factor of $x^n+c^n$when $n\ge 1$