Question

$x^n-y^n$ is divisible by $x+y$, when:

• A. $n$ is rational
• B. $n$ is an positive integer
• C. $n$ is an even positive integer
• D. $n$ is an odd

Answer 1

Answer: (C)

$n$ is an even positive integer

Let,
$f\left(x\right)=x^n-y^n$ and
$g\left(x\right)=x+y$
$=>x+y=0$
$=>x=-y$
If $f\left(x\right)$ is divided by $g\left(x\right)$ then,$f(-y)$ should be equal to zero.
Therefore,
$f(-y)=(-y{)}^n-y^n$
$=0$ if $n$ is even
$\ne 0$ if $n$ is odd
Therefore,
$x+y$ is a factor of $x^n-y^n$ when $n$ is a even integer.