Chapter 4.2, Problem 91AYU

To determine

To prove: Using factor theorem, $\left(x-c\right)$ is a factor of $x^n-c^n$ for any positive integer $n$.

Answer to Problem 91AYU

$\text{Using factor theorem, }(x-c)\text{ is a factor of }{x}^n-c^n\text{ for any positive integer }n\text{ is proved}$.

Explanation of Solution

Given:

Let $f\left(x\right)=x^n-c^n$ be any polynomial function.

Divisor $=\left(x-c\right)$

The factor theorem states that if $f\left(c\right)=0$, then $\left(x-c\right)$ is a factor of $f\left(x\right)=x^n-c^n$

Putting $\left(x=c\right)$ in $f\left(x\right)=x^n-c^n$ and result $=0$, then $\left(x-c\right)$ is the factor of $f\left(x\right)=x^n-c^n$

$f\left(c\right)=c^n-c^n=0$where value of $n\ge 0$