Question

Find the value of b for which the polynomial $2x^3+9x^2-x-b$ is exactly divisible by $2x+3$?

• A. $15$
• B. $-15$
• C. $10$
• D. $-10$

Answer 1

The correct option is A $15$

Given that, $2x+3$ is a factor of the polynomial $p\left(x\right)=2x^3+9x^2-x-b$
According to the factor theorem, if $(x-a)$ is a factor of $f\left(x\right)$, then $f\left(a\right)=0$.

Therefore, by Factor theorem $p(-\frac{3}{2})=0$

$\Rightarrow 2{(-\frac{3}{2})}^3+9{(-\frac{3}{2})}^2-(-\frac{3}{2})-b=0$

$\Rightarrow -\frac{27}{4}+\frac{81}{4}+\frac{3}{2}-b=0$

$\Rightarrow \frac{-27+81+6}{4}-b=0$

$\Rightarrow b=\frac{60}{4}=15$

$\therefore b=15$

Hence, option A is correct.