Question

For what value of a is the polynomial (2x³ + ax² + 11x + a + 3) exactly divisible by (2x − 1)?

Answer 1

Let:
$f\left(x\right)=2x^3+a{x}^2+11x+a+3$
Now,
$2x-1=0\Rightarrow x=\frac{1}{2}$
By the factor theorem, $f\left(x\right)\text{is exactly divisible by}\left(2x-1\right)\text{if}f\left(\frac{1}{2}\right)=0$.
Thus, we have:
$$\begin{gathered} f\left(\frac{1}{2}\right)=\left[2\times {\left(\frac{1}{2}\right)}^3+a\times {\left(\frac{1}{2}\right)}^2+11\times \left(\frac{1}{2}\right)+a+3\right] \\ =\left(\frac{1}{4}+\frac{a}{4}+\frac{11}{2}+a+3\right) \\ =\left(\frac{35}{4}+\frac{5a}{4}\right) \end{gathered}$$
Also,
$$\begin{gathered} f\left(\frac{1}{2}\right)=0 \\ \Rightarrow \frac{35}{4}+\frac{5a}{4}=0 \\ \Rightarrow \frac{5a}{4}=-\frac{35}{4} \\ \Rightarrow a=-7 \end{gathered}$$

Hence, $f\left(x\right)\text{is exactly divisible by}\left(2x-1\right)\text{when}a\mathrm{is}-7.$