Question

What is the method for factoring -x^3+3x+2?

Answer 1

One way is to find a root, then factor out that term. We can make an educated guess using the Rational Root Theorem, but however we do it, notice that
-2^3+3(2)+2 = -8+6+2 = 0,
so x=2 is a root. That means the term (x-2) factors out. Do you know how to divide polynomials? Because now we want to divide -x^3+3x+2 by x-2 to get what the other terms are. What's left is a quadratic, so we can factor that just fine.

Answer 2

Is there a systematic way of solving it?

Answer 3

I would say that is the systematic way..try to factor out (x +- 1) or (x+- 2) using polynomial division or synthetic division.

Answer 4

I don't think I understand. Can you show me with this equation?

Answer 5

somethingawesome already showed you that (x-2) does indeed factor out evenly.
But the way to approach it is there are only 4 possible terms which will factor out evenly... (x +- 1) or (x +- 2).
do you know how to do synthetic division?

Answer 6

If you want an equation like the quadratic one but for cubics, there are some here:
http://en.wikipedia.org/wiki/Cubic_function
I don't think they'll be very helpful though.

Answer 7

2| -1 0 3 2
-2 -4 -2
_______________
-1 -2 -1 0

This is an example of dividing by (x-2) using synthetic division.
The last number is zero which is the remainder, thus it divides evenly.
The other numbers are the coefficients in the quotient
Factored form:
(x-2)(-x^2 -2x -1)
Pull out a neg
-(x-2)(x^2 +2x +1)
factor quadratic...notice its a perfect square
-(x-2)(x+1)^2

Answer 8

Got it. Thanks!

Answer 9

ok do you understand the synthetic division?

Answer 10

I can do regular polynomial divsion, and I'm looking up how to do synthetic division on purplemath. I guess if worst comes to worst I can always use long division

Answer 11

no either way is fine..synthetic is just easier at times especially in these cases
the goal is just to find a root and divide it out to get it into factored form.
you don't need to spend too much time learning it if you are comfortable with regular division