Question

Let $P\left(x\right)$ be a real polynomial of degree $3$ which vanishes at $x=–3.$ Let $P\left(x\right)$ have local minima at $x=1,$ local maxima at $x=–1$ and $\underset{-1}{\overset{1}{\int }}P\left(x\right)dx=18$, then the sum of all the coefficients of the polynomial $P\left(x\right)$ is equal to

Answer 1

From the given conditions, we can write
$P^{\prime }\left(x\right)=a(x+1)(x-1)\therefore P\left(x\right)=\frac{a{x}^3}{3}-ax+C$
We know
$P(-3)=0$
$\Rightarrow a(-9+3)+C=0$
$\Rightarrow 6a=C\cdots \left(i\right)$
Also,
$\underset{-1}{\overset{1}{\int }}P\left(x\right)dx=18\Rightarrow \underset{-1}{\overset{1}{\int }}(\frac{a{x}^3}{3}-ax+C)dx=18\Rightarrow 2C=18\Rightarrow C=9$
From $\left(i\right)$
$a=\frac{3}{2}\therefore P\left(x\right)=\frac{x^2}{2}-\frac{3}{2}x+9$

Hence, the sum of coefficients
$=-1+9=8$