Question

Determine which of the following polynomials has $(x+1)$ a factor:
(i) $x^3+x^2+x+1$
(ii) $x^4+x^3+x^2+x+1$
(iii) $x^4+3x^3+3x^2+x+1$
(iv) $x^3-x^2-(2+\sqrt{2})x+\sqrt{2}$

Answer 1

Apply remainder theorem

$x+1=0$

$x=-1$

Put the value of $x=-1$ in all equations.

(i) $x^3+x^2+x+1=(-1{)}^3+(-1{)}^2+(-1)+1=-1+1-1+1=0$

Then $x+1$ is the factor of the equation.

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

This is not zero. Then $x+1$ is not the factor of the equation

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

This is not zero. Then $x+1$ is not the factor of equation.

(iv) $x^3-x^2-(2+\sqrt{2})x+\sqrt{2}$

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

$=-1-1+2+\sqrt{2}+\sqrt{2}=2\sqrt{2}$

This is not zero. Then $x+1$ is not the factor of the equation.