Factoring Examples

Date: 12/16/96 at 21:31:31
From: Leslie McClendon
Subject: Algebra-factoring completely

Dear Dr Math, 

There are no other Web sites that can answer my questions and explain 
them step by step. Here are my last few questions:

1. Factor Completely   x(x+1)(x-4) + 4(x+1)

2. Factor Completely   a(a^2-9) - 2(a-3)^2

3. Factor Completely   x^4 - x^2 + 4x - 4

4. Factor Completely   t^4 - 10t^2 + 9

I'm sure that if you answer these four questions and explain them, I 
will be able to figure out the rest of my work. 

Thanks, Leslie

Date: 12/17/96 at 13:58:09
From: Doctor Tom
Subject: Re: Algebra-factoring completely

Hi Leslie,

It looks like you've already learned a lot about factoring in your 
class because you're now looking at some problems that require a 
little thought.  They use techniques you already know, but many times 
you won't get to the answer in a single step.  Let's look at your 
examples:

Example 1:

   x(x+1)(x-4) + 4(x+1)

I see (x+1) in both terms, so I'll begin by factoring it out:

   (x+1)[x(x-4) + 4]

The thing in brackets is a mess, so I'll multiply it out:

   (x+1)[x^2 - 4x + 4]

But the thing in brackets can now be factored in the usual way:

   (x+1)(x-2)(x-2)

Example 2:

   a(a^2-9)-2(a-3)^2

I notice that a^2-9 is (a+3)(a-3), which is nice because there's an 
(a-3) in the other term:

   a(a+3)(a-3) - 2(a-3)^2

   = (a-3)[a(a+3) - 2(a-3)]

Now multiply out the junk in the brackets:

   = (a-3)[a^2 + 3a - 2a + 6]

   = (a-3)[a^2 + a + 6]

The thing in brackets can't be factored, so you're done.

Example 3:

   x^4 - x^2 + 4x - 4

I notice that if I let x = 1, this is zero, so I know that (x-1)
is a factor:

   x^2(x^2 - 1) + 4(x-1)

   = x^2(x+1)(x-1) + 4(x-1)

   = (x-1)[x^2(x+1) + 4]

   = (x-1)[x^3 + x^2 + 4]

Notice that if I put x=-2 in the expression in brackets, it will be 
zero, so x+2 is a factor:

   = (x-1)(x+2)(x^2 -x + 2)

And that's as far as it goes.

Example 4:

   t^4 - 10t^2 + 9

Suppose u = t^2.  Then this looks like u^2 - 10u + 9.  Could
you factor that?  Of course:

   = (u - 9)(u - 1)

But u = t^2, so it's really:

   = (t^2 - 9)(t^2 - 1)

And both terms factor:

   = (t+3)(t-3)(t+1)(t-1)

Notice I've used a bunch of different tricks.  You should get familiar
with them.  There's more than one way to solve an algebra problem!

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/