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 ∫_{-1}^{1} P(x) dx = 18, then the sum of all the coefficients of the polynomial P(x) is equal to ________.

Hint:

The sum of all coefficients of a polynomial $P(x)$ is simply $P(1)$.

Answer 1

Correct Answer: 8

Step 1: $P'(x)$ must have roots at $1$ and $-1$. So $P'(x) = k(x-1)(x+1) = k(x^2 - 1)$.
Step 2: Integrate $P'(x)$: $P(x) = k(\frac{x^3}{3} - x) + C$.
Step 3: Given $P(-3) = 0$: $k(-9 + 3) + C = 0 \implies C = 6k$. So $P(x) = k(\frac{x^3}{3} - x + 6)$.
Step 4: Given $\int_{-1}^{1} P(x) dx = 18$: $k \int_{-1}^{1} (\frac{x^3}{3} - x + 6) dx = k [0 + 0 + 6(1 - (-1))] = 12k = 18 \implies k = 1.5$.
Step 5: $P(x) = \frac{3}{2}(\frac{x^3}{3} - x + 6) = \frac{1}{2}x^3 - \frac{3}{2}x + 9$.
Step 6: Sum of coefficients $= P(1) = 1/2 - 3/2 + 9 = -1 + 9 = 8$.