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Factoring Cubic Polynomials

Date: 01/01/2002 at 00:22:38
From: Ash
Subject: Factoring Cubic equations

Hello Dr. Math,

I was hoping that you could help me figure out a few problems on 
factoring. I know how to factor square polynomial equations, by using 
the quadratic formula. But I have a lot of problems factoring cubic 
polynomials. Would you be able to help me on a few of these problems? 

1) 10*x^3 - 3*x^2 - x = f(x)

2) 18*r^3 - 128*r^5*s^2

By the way, what does it mean to factor a polynomial into "linear 
factors"? Is that the same as regular factoring?


Date: 01/01/2002 at 02:38:05
From: Doctor Paul
Subject: Re: Factoring Cubic equations

Factoring into linear factors means that each of the factors should be 
linear (i.e., of degree one).

The first problem is a quadratic and you can apply the quadratic 
formula. Factor out an x from everything and then apply the quadratic 
formula to what's left behind.

In the second problem, factor out the largest thing possible: 2*r^3

That gives:

   [2*r^3] * (9 - 64r^2*x^2)

The second part is a difference of squares so it factors as well. Can 
you finish this?

In general, factoring cubics is hard. The only way you can be expected 
to be able to factor cubics is if they are "special" cubics. By 
special, I mean cubics that are chosen in such a way that they factor 
nicely in ways with which you are already familiar. As a general rule 
for factoring cubics, try to factor out the largest thing possible and 
then see what's left behind. If you're lucky you can apply a readily 
available factoring technique to what's left behind and then you'll be 
finished with the problem.

I hope this helps. Please write back if you'd like to talk about this 
some more.

- Doctor Paul, The Math Forum   
Associated Topics:
High School Polynomials

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