Question

If $p\left(x\right)=2x^3-11x^2-4x+5\mathrm{and}g\left(x\right)=2x+1$, show that g(x) is not a factor of p(x).

Answer 1

$p\left(x\right)=2x^3-11x^2-4x+5$

$g\left(x\right)=2x+1=2\left(x+\frac{1}{2}\right)=2\left[x-\left(-\frac{1}{2}\right)\right]$

Putting $x=-\frac{1}{2}$ in p(x), we get

$p\left(-\frac{1}{2}\right)=2\times {\left(-\frac{1}{2}\right)}^3-11\times {\left(-\frac{1}{2}\right)}^2-4\times \left(-\frac{1}{2}\right)+5=-\frac{1}{4}-\frac{11}{4}+2+5$ $=-\frac{12}{4}+7=-3+7=4\ne 0$

Therefore, by factor theorem, (2x + 1) is not a factor of p(x).

Hence, g(x) is not a factor of p(x).