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 d x = 18 , then the sum of all the coefficients of the polynomial P ( x ) is equ

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 &#8747; - 1 1 P x d x = 18 , then the sum of all the coefficients of the polynomial P ( x ) is equal to ___ .

Quantrex Academy · Accepted Answer

Correct answer: 0

Let P ' ( x ) = a ( x - 1 ) ( x + 1 ) = a x 2 - 1 P x = a &#8747; x 2 - 1 d x + c = a x 3 3 - x + c Now P ( - 3 ) = 0 &#8658; a - 27 3 + 3 + c = 0 &#8658; - 6 a + c = 0 &#160;&#160; &#8230; ( 1 ) Now &#8747; - 1 1 a x 3 3 - x + c d x = 18 = 2 c = 18 &#8658; c = 9 &#160; &#160; . . . 2 From ( 1 ) &#160; &#38; &#160; ( 2 ) &#8658; - 6 a + 9 = 0 &#8658; a = 3 2 &#8658; P x = 3 2 x 3 3 - x + 9 Then, the Sum of coefficient = 1 2 - 3 2 + 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 ∫ - 1 1 P x d x = 18 , then the sum of all the coefficients of the polynomial P ( x ) is equal to ___ .

Correct answer

Step-by-step solution

More from Application of Derivatives

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 &#8747; - 1 1 P x d x = 18 , then the sum of all the coefficients of the polynomial P ( x ) is equal to ___ .

Correct answer

Step-by-step solution

More from Application of Derivatives

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 d x = 18 , then the sum of all the coefficients of the polynomial P ( x ) is equal to ___ .