"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 $\int\limits_{ - 1}^1 {P(x)dx}$ = 18, then the sum of all the coefficients of the polynomial P(x) is equal to _________."

Question

Accepted Answer

"P'(x) = a(x + 1)(x $-$ 1)

$\therefore$ P(x) = ${{a{x^3}} \over 3}$ $-$ ax + C

P($-$3) = 0 (given)

$\Rightarrow$ a($-$9 + 3) + C = 0

$\Rightarrow$ 6a = C ..... (i)

Also, $\int\limits_{ - 1}^1 {P(x)dx} = 18$

$$\Rightarrow \int\limits_{ - 1}^1 {\left( {a\left( {{{{x^3}} \over 3} - x} \right) + C} \right)} dx = 18$$

$\Rightarrow 0 + 2C = 18 \Rightarrow C = 9$

from (i)

$a = {3 \over 2}$

$\therefore$ $P(x) = {{{x^3}} \over 2} - {3 \over 2}x + 9$

Sum of co-efficient

= ${1 \over 2} - {3 \over 2} + 9$ = $-$1 + 9 = 8"

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

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

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