Question

In the following case, use factor theorem to find whether $g\left(x\right)$ is a factor of the polynomial $p\left(x\right)$ or not.
$p\left(x\right)=3x^4+3x^2-4x-11g\left(x\right)=x-\frac{1}{2}$

Answer 1

The Factor Theorem states that if $a$ is the root of any polynomial $p\left(x\right)$ that is if $p\left(a\right)=0$, then $(x-a)$ is the factor of the polynomial $p\left(x\right)$.

It is given that $p\left(x\right)=3x^4+3x^2-4x-11$ and $g\left(x\right)=x-\frac{1}{2}$, therefore, by factor theorem $(x-\frac{1}{2})$ is the factor of $p\left(x\right)$ if $p\left(\frac{1}{2}\right)=0$ and thus,

$p\left(\frac{1}{2}\right)=3{\left(\frac{1}{2}\right)}^4+3{\left(\frac{1}{2}\right)}^2-(4\times \frac{1}{2})-11=(3\times \frac{1}{16})+(3\times \frac{1}{4})-2-11=\frac{3}{16}+\frac{3}{4}-13=\frac{3+12-208}{16}=\frac{15-208}{16}=-\frac{193}{16}$

$\ne 0$

Since $p\left(\frac{1}{2}\right)$$\ne 0$

Hence, $(x-\frac{1}{2})$ is not a factor of $p\left(x\right)=3x^4+3x^2-4x-11$.