f(x) = x^3
f(-2) = -8
The text says that by dividing a function [x^3] by [x minus a given input (-2)] using synthetic division, I'll be able to produce the correct output (-8). I want to know why this happens.
(x^3)/(x-(-2)) = -8
The Attempt at a Solution
I'm trying to prove that x^3 divided by x minus the input will produce the output. I made it simple enough, but all I manage to do is prove that synthetic division does what it does.
(x^3)/(x+2) = -8 [multiply both sides by (x+2)]
x^3 = -8x - 16 [move right side to left]
x^3 + 8x + 16 [divide whole equation by (x+2)]
(x^3 + 8x + 16)/(x+2) [synthetic division]
-2 1 0 8 16
-2 4 -24
1 -2 4 -8
I still don't know why synthetic division produces an output for a given input when subtracted from x, and becomes the divisor for the original equation. It still ends up with me having to use synthetic division to obtain my result.