Find the coefficient of the squared term

Question

Find the coefficient of the squared term in the simplified form for the second derivative, f "(x) for f(x) = (x³ + 2x + 3)(3x³ - 6x² - 8x + 1). Use the hyphen symbol, -, for negative values. 

Answer 1

In the second derivative of a polynomial which f(x) is in this case, the quadratic term increases by one when we integrate to get the first derivative, f'(x), and then increases by one again when we integrate to get the function, f(x), back again. So the quadratic term of the second derivative of a polynomial originated before differentiating with the quartic, 4th degree term in f(x).

Looking at the products of the terms of the factors of f(x), (x³ + 2x + 3) and (3x³ - 6x² - 8x + 1), we can see that we will get a 4th power term, when we multiply x³ and - 8x together and when we multiply 2x and 3x³ together. The sum of those two products is -8x⁴ + 6x⁴ = -2x⁴.

(-2x^4)' = -8x³

(-2x^4)'' = (-8x³)' = -24x²

So, the coefficient of the squared term in f''(x) originating from the 4th power term in f(x), is -24