Question:
If $p(x)=2 x^{5}-11 x^{2}-4 x+5$ and $g(x)=2 x+1$, show that $g(x)$ is not a factor of $p(x)$.
Solution:
$p(x)=2 x^{3}-11 x^{2}-4 x+5$
$g(x)=2 x+1=2\left(x+\frac{1}{2}\right)=2\left[x-\left(-\frac{1}{2}\right)\right]$
Putting $x=-\frac{1}{2}$ in $p(x)$, we get
$p\left(-\frac{1}{2}\right)=2 \times\left(-\frac{1}{2}\right)^{3}-11 \times\left(-\frac{1}{2}\right)^{2}-4 \times\left(-\frac{1}{2}\right)+5=-\frac{1}{4}-\frac{11}{4}+2+5=-\frac{12}{4}+7=-3+7=4 \neq 0$
Therefore, by factor theorem, (2x + 1) is not a factor of p(x).
Hence, g(x) is not a factor of p(x).