Question

(x+1) is a factor of $x^n$ + 1 only if

• A. n is an odd integer
• B. n is an even integer
• C. n is a negative integer
• D. n is a positive integer

Answer 1

The correct option is A

n is an odd integer

$\therefore$ (x+1) is a factor of $x^n$ + 1

Let x + 1 = 0, then x =-1

$\therefore$ f(x) = $x^n$ + 1

and f(-1) $=(-1{)}^n$ + 1

But $(-1{)}^n$ is positive if n is an even integer and negative if n is an odd integer

and $(-1{)}^n$ + 1 = 0

{$\because$ x + 1 is a factor of f(x)}

$\therefore (-1{)}^n$ must be negative

$\therefore$ n is an odd integer