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/
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