Question

Question 1
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

To have (x + 1) as a factor, substituting x = - 1 must give p(-1) = 0.
(i) $x^3+x^2+x+1=(-1{)}^3+(-1{)}^2+(-1)+1=-1+1-1+1=0$
Therefore, $x+1$ is a factor $x^3+x^2+x+1$.

(ii) $x^4+x^3+x^2+x+1=(-1{)}^4+(-1{)}^3+(-1{)}^2+(-1)+1=1-1+1-1+1=1$
Remainder is not 0. Therefore (x + 1) is not its factor.

(iii) $x^4+3x^3+3x^2+x+1=(-1{)}^4+3(-1{)}^3+3(-1{)}^2+(-1)+1=1-3+3-1+1=1.$ Remainder is not 0.
Therefore, (x + 1) is not a factor.
(iv) $x^3+x^2-(2+\sqrt{2})x+\sqrt{2}=(-1{)}^3-(-1{)}^2-(2+\sqrt{2})(-1)+\sqrt{2}=-1-1+2+\sqrt{2}+\sqrt{2}=2\sqrt{2}$
Remainder not 0.
Therefore, (x + 1) is not a factor.