Question

Determine which if the following polynomials has (x+1) a factor
$x^{4}+x^{3} +x^{2} +x+1$

Answer 1

Given :-

$x^{4}+x^{3} +x^{2} +x+1$

To determine :-

If ( x + 1) is the factor of given polynomial or not.

Solution:-

If (x + 1) is the factor of given polynomial then it must satisfies it

I. e, p (x) = 0

$x + 1 = 0$

$x = -1$

Putting the value of x in given polynomial.

$\implies$ $p(x) = {x}^{4} + {x}^{3} + {x}^{2} + x + 1$

$\implies$ $p( - 1) = {( - 1)}^{4} + {( - 1)}^{3} + {( - 1)}^{2} + ( - 1) + 1$

$\implies$ $p( - 1) = 1 - 1 + 1 - 1 + 1$

$\implies$ $p( - 1) = 3 - 2$

$\implies$ $p( - 1) = 1$

hence, it doesn't satisfy the p(x)

it doesn't satisfy the p(x) Therefore, ( x + 1) is not the factor of given polynomial.