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## Homework Statement

f(x) = x^3

Find f(-2)

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.

## Homework Equations

(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.