Factoring Cubic Expressions

Date: 08/24/2010 at 18:25:50
From: katy
Subject: factoring

I don't know how to factor numbers with variables to the third power. For
example, how do I factor this?

   16x^3 - 250

Date: 08/25/2010 at 10:28:34
From: Doctor Ian
Subject: Re: factoring

Hi Katy,

The first thing I'd do is look for other cubes.

For example, I can write your cubic expression this way:

     2(8x^3 - 125)

   = 2(2^3 x^3 - 5^3)

   = 2((2x)^3 - 5^3)

What might you try next?

Now, if you're not particularly facile with cubes, you could make things a
little more visible by finding prime factors, and expanding exponents. For
example,

     16x^3 - 250

   = 2*2*2*2*x*x*x - 2*5*5*5
       |___| |___|     |___|
         |_____|_________|______  Here are the cubes!

In some ways, prime factors are a kind of x-ray machine for uncovering
hidden structure in expressions.  :^D

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/