SYNTHETIC DIVISION problem...?

Question

I'm having trouble with this:

http://img107.imageshack.us/img107/4207/664vq2.png

I know how to use synthetic, but how do I put it in its factored form? Any help will be appreciated.

Answer 1

You are told that you have a *double* root at x = 2. This is the same as x - 2 = 0

You are also told you have a single root at x = -1. This is the same as x + 1 = 0

So multiply it out:
f(x) = (x - 2)(x - 2)(x + 1)
f(x) = (x² - 4x + 4)(x + 1)
f(x) = (x^3 - 4x² + 4x) + (x² - 4x + 4)
f(x) = x^3 - 3x² + 4

Okay, but now the problem is we need the y-intercept to be:
-3.2.

If I plug in x = 0, I'm supposed to get -3.2, but I get 4.
So multiply the whole equation by -3.2/4 = -0.8
P(x) = -0.8 * f(x)
P(x) = -0.8(x^3 - 3x² + 4)
P(x) = -0.8x^3 + 3(0.8)x² - 4(0.8)

The final equation is:
P(x) = -0.8x^3 + 2.4x² - 3.2

As a double-check, I graphed this equation and confirmed that it has a double root at x = 2, a single root at x = -1 and a y-intercept of -3.2.

Answer 2

I don't see the need for synthetic division... you know that there are 2 roots at x=2 and one at x=-1... that means that you have the multiply (x-2)(x-2)(x-(-1))... this multiplies to x^3-3x^2-2x+2... that equation has intercepts at 2,2 and -1... but wait! It only has a y intercept of 2... A nice thing about these equations is that you can multiply it by any number and It'll still have the same roots... sooo.... we currently have an intercept of 2... but we want it to be -3.2... The way to fix this is to multiply the WHOLE equation by -1.6
so -1.6(x^3-3x^2-2x+2) has the specified intercepts... {2,2,-1} and the y int of -3.2