Question

Let $P ( x )$ be a real polynomial of degree 3 which vanishes at $x=-3 .$ Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and $\int\limits_{-1}^{1} P ( x ) d x =18$, then the sum of all the coefficients of the polynomial $P ( x )$ is equal to ________.

Answer 1

Let $p'(x)=a(x-1)(x+1)=a\left(x^{2}-1\right)$
$p(x)=a \int\left(x^{2}-1\right) d x+c$
$=a\left(\frac{x^{3}}{3}-x\right)+c$
Now $p (-3)=0$
$\Rightarrow a \left(-\frac{27}{3}+3\right)+ c =0$
$\Rightarrow -6 a + c =0 \quad \ldots(1)$
Now $\int\limits_{-1}^{1}\left( a \left(\frac{ x ^{3}}{3}- x \right)+ c \right) d x \mid=18$
$=2 c =18 \Rightarrow c =9 \ldots$(2)
$\Rightarrow $ from $(1) \&(2)$
$ \Rightarrow -6 a+9=0$
$ \Rightarrow a=\frac{3}{2}$
$\Rightarrow p(x)=\frac{3}{2}\left(\frac{x^{3}}{3}-x\right)+9$
sum of coefficient
$=\frac{1}{2}-\frac{3}{2}+9$
$=8$