Thread: Find a cubic polynomial P for which...

1. Find a cubic polynomial P for which...

Apostol, 1.26.24

Problem: Find a cubic polynomial P for which P(0) = P(-2) = 0, P(1) = 15, and the integral of P from -2 to 0 is 4/3.

I can find a polynomial that satisfies the first 3 conditions with no problem:
x(x + 2)(x + c)...
(1)((1) + 2)((1) + c) = 15
x^3 + 6x^2 + 8x

However, I calculated the integral of x^3 + 6x^2 + 8x from -2 to 0 to be 0. Thus, I can't scale it with a constant, and I don't know of any other transformations that would allow me to preserve the intercepts while setting the area to 4/3.

I don't need the answer -- it's in the book. But any advice on getting from here to there would be much appreciated.

2. Originally Posted by Dementiy
Apostol, 1.26.24

Problem: Find a cubic polynomial P for which P(0) = P(-2) = 0, P(1) = 15, and the integral of P from -2 to 0 is 4/3.

I can find a polynomial that satisfies the first 3 conditions with no problem:
x(x + 2)(x + c)...
(1)((1) + 2)((1) + c) = 15
x^3 + 6x^2 + 8x

However, I calculated the integral of x^3 + 6x^2 + 8x from -2 to 0 to be 0. Thus, I can't scale it with a costant, and I don't know of any other transformations that would allow me to preserve the intercepts while setting the area to 4/3.

I don't need the answer -- it's in the book. But any advice on getting from here to there would be much appreciated.
Try setting your polynomial as

$\displaystyle P(x) = kx(x + 2)(x + c)$ instead.