Question

determine which of the following polynomials has ( x+1) a factor:a x^3+x^2 +x+1 , x^4+x^3 +x^2+ x+1

Answer 1

xpower3+xpower2+x+1 has x+1 as factor because by dividing them we get reminder zero

Answer 2

x-1 = 0

x = -1

$f( - 1) = x {}^{3} + x {}^{2} + x + 1$

$f( - 1) = ( - 1) {}^{3} + ( - 1) {}^{2} + ( - 1) + 1$

$= - 1 + 1 - 1 + 1$

$f( - 1) = 0$

So (x + 1) is a factor of above polynomial.

$f( - 1) = x {}^{4} + x {}^{3} + x {}^{2} + x + 1$

$f( - 1) = ( - 1) {}^{4} + ( - 1) {}^{3} + ( - 1) {}^{2} + ( - 1) + 1$

$= 1 - 1 + 1 - 1 + 1$

$f( - 1) = 1$

Hence (x + 1) is not a factor of above polynomial.