Find a formula for a cubic function f if f(5) = 240 and f(−3) = f(0) = f(6) = 0.

Answer 1

Bradford's method works whenever we are solving for a cubic and have the value of the function at four points. We can take advantage of the fact that we have 3 0s to solve in a different manner in this case.

As f(-3) = f(0) = f(6) = 0, the function will be of the form C(x+3)(x)(x-6) =

As f(5) = 240, C(5+3)(5)(5-6) = -40C

-40C = 240

C = -6

f(x) = -6(x+3)(x)(x-6) = -6(x³ - 3x² - 18x) = -6x³ + 18x² + 108x

Answer 2

I'm not sure how this was answered before, but need to fit the form

f(x) = Ax³+Bx²+Cx+D

There are 4 equations and 4 unknowns. However f(0) = 0 immediately make D=0. So we have 3 equations and 3 unknowns:

[125 25 5][A] = [240]

[-27 9 -3][B] = [0]

[216 36 6][C] = [0]

In Reduced Row Echelon Form:

[1 0 0 | -6]

[0 1 0 | 18]

[0 0 1 |108]

Gives:

f(x) = -6x³+18x²+108x