Question

What must be added to the polynomial $p\left(x\right)=2x^3+9x^2-5x-15$, so that the resulting polynomial is exactly divisible by $2x+3$.

Answer 1

$-6$ must be added to the polynomial $2x^3+9x^2-5x-15$ so that the resulting polynomial is exactly divisible by $2x+3$

Step by step explanation :

Given polynomial is $P\left(X\right)=2x^3+9x^2-5x-15$

Let a is the number which must be added to above polynomial so that it is exactly divisible by $2x+3$

So, by remainder theorem

$P\left(\frac{-3}{2}\right)+a=0$

$\Rightarrow 2{\left(\frac{-3}{2}\right)}^3+9{\left(\frac{-3}{2}\right)}^2-5\left(\frac{-3}{2}\right)-15+a=0$

$\Rightarrow \frac{-27}{4}+\frac{81}{4}+\frac{-15}{2}+a=0$

$\Rightarrow 6+a=0\Rightarrow a=-6$

Hence, $-6$ must be added to the polynomial $2x^3+9x^2-5x-15$ so that the resulting polynomial is exactly divisible by $2x+3$