Question

Determine which of the following polynomial has $(x+1)$ a factor : $x^3–x^2–(2+\sqrt{2})x+\sqrt{2}$

Answer 1

Determining whether polynomial has $(x+1)$ a factor or not

Let $p\left(x\right)=x^3-x^2-(2+\sqrt{2})x+\sqrt{2}$ be the polynomial

By factor theorem

If $p\left(x\right)$ is a polynomial of degree $n>1$ and $a$ is any real number, then $x–a$ is a factor of $p\left(x\right)$, if $p\left(a\right)=0$,

Therefore, $(x+1)$ is a factor of $p\left(x\right)$, if $p(-1)=0$

$p\left(x\right)=x^3-x^2-(2+\sqrt{2})x+\sqrt{2}$

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

$p(-1)=-1-1+\left(2+\sqrt{2}\right)+\sqrt{2}$

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

$p(-1)=2\sqrt{2}$

Since, $p(-1)\ne 0$

Hence, $(x+1)$ is not a factor of polynomial $x^3-x^2-(2+\sqrt{2})x+\sqrt{2}$.