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 ${\int }_{-1}^{1}P\left(x\right)dx=18$, then the sum of all the coefficients of the polynomial $P\left(x\right)$ is equal to

Answer 1

The correct answer is $8$

Explanation for the correct answer.

Finding the value of the sum of all the coefficients of the polynomial $P\left(x\right)$ is equal to

Given: $P\left(x\right)$ a real polynomial of degree $3$ which vanishes at $x=-3.$ Also $P\left(x\right)$ has local minima at $x=1$, local maxima at$x=-1$ and ${\int }_{-1}^{1}P\left(x\right)dx=18$,

Let, $P\text{'}\left(x\right)=a(x+1)(x–1)$ then

$$\begin{gathered} P\left(x\right)=\left(\frac{a{x}^3}{3}\right)–ax+c \\ P(-3)=0 \\ \Rightarrow a(-9+3)+C=0 \\ \Rightarrow 6a=C\dots \left(i\right) \end{gathered}$$

Also,$$\begin{gathered} {\int }_{-1}^{1}P\left(x\right)dx=18 \\ \Rightarrow {\int }_{-1}^1[a\left(\left(\frac{x^3}{3}\right)–x\right)+c]dx=18 \end{gathered}$$

$$\begin{gathered} \Rightarrow 0+2C=18 \\ \Rightarrow C=9 \end{gathered}$$

From $\left(i\right),a=\frac{3}{2}$

$P\left(x\right)=\left(\frac{x^3}{2}–\frac{3}{2}\right)x+9$

Sum of co-efficient $=–1+9=8$

Hence the value of the sum of all the coefficients of the polynomial $P\left(x\right)$ is equal to $8$