Let P(x) be a real polynomial of degree 3 which vanishes at x=-3.

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 $\displaystyle\int_{-1}^1P(x)\,dx=18$. Then the sum of all the coefficients of the polynomial $P(x)$ is equal to

Answer: 8

Accepted Answer

Since the local extrema are at $x=\pm1$: $P'(x)=a(x-1)(x+1)=a(x^2-1)$, so $P(x)=a\left(\dfrac{x^3}3-x\right)+C$.

$P(-3)=0$: $a\left(-9+3\right)+C=0 \implies -6a+C=0$.

$\displaystyle\int_{-1}^1P(x)\,dx = 2C = 18 \implies C=9$ (the odd part integrates to zero).

From $-6a+9=0$: $a=\dfrac32$.

$$P(x) = \frac32\left(\frac{x^3}3-x\right)+9$$

Sum of coefficients $= P(1) = \dfrac32\left(\dfrac13-1\right)+9 = -1+9 = 8$

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