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Why Factor Polynomials?

Date: 02/20/98 at 18:52:05
From: Alexandra Wrigley
Subject: Factoring polynomials

Why is factoring polynomials important - how do we use it in our 
everyday lives?

Date: 02/21/98 at 13:42:31
From: Doctor Sam
Subject: Re: Factoring polynomials


Well, factoring ISN'T important to most people in everyday life. I 
mean shopping and cleaning and cooking and going to the movies. But 
many occupations use different kinds of mathematics, ranging from 
accountants to carpenters to scientists and engineers to people who 
work to protect the environment.

Many of them will sometimes need to use factoring, but factoring isn't 
a goal in itself. Factoring is used to solve different kinds of 
problems. I think you might want to know why you should learn 
factoring if you aren't ever going to use it "in real life."  

One answer to your question is that most of us don't know what we will 
be doing in real life until it happens to us. Sometimes we plan for it 
and sometimes it takes us by surprise. But it is a good idea to be 
prepared. If you don't know ANY mathematics then there are hundreds, 
maybe thousands of jobs that you won't be able to do. For most of 
these jobs mathematics isn't the main point of the job, it is just one 
of the many tools that are used. So if you don't know mathematics you 
may be losing the opportunity to do something that you would find 
exciting and worthwhile.

And now to the mathematical part of your question: how is factoring 

I can think of two important uses of factoring.  

One is to make complicated things look simpler. For example,

   I don't have any idea what this fraction 1848 / 16632 means.  
   The numbers are too large.  But if I factor the numerator and    
   denominator I get:  (3)(8)(7)(11) / (8)(11)(7)(9) and I can 
   see that the 3, the 8, the 7, and a 5 are factors of both the  
   numerator and denominator.  Since 3/3, 8/8, 7/7 and 11/11 are all  
   equal to 1 the fraction reduces to 1/3, which is a lot easier to  
   understand and to compute with.

It's the same with complex algebraic fractions. (x^4 - x^2)/(x^3+x^2) 
looks REALLY complicated, but the numerator factors into x^2(x-1)(x+1) 
and the denominator factors into x^2(x+1), and since x^2/x^2 and 
(x+1)/(x+1) are both equal to 1, this fraction simplifies to x-1 . . . 
a LOT easier to work with.

A second important use of factoring is in solving equations. You don't 
need to factor to solve 2x+3 = 5 ... linear equations use a different 
method. And you don't need to factor second degree equations because 
you can use the Quadratic Formula (although factoring is often MUCH 
easier!). But if you need to solve equations where the degree of the 
highest term is more than 2 then you really have no choice at all 
because you don't have formulas for most of them.

Here is an example of a hard equation to solve:  

   x^5 - 4x^4 - 12x^3 = 0

But if you factor it completely you get (x^3)(x-6)(x+2) = 0 and now it 
is easy, because this says that a product of things turns out to be 
equal to zero. If you multiply, the only way to get zero as an answer 
would be if you multiplied by zero. So one of the three factors has 
to be zero.

    If x^3 = 0 then x = 0
    If x-6 = 0 then x = 6
    If x+2 = 0 then x = -2

So the solutions to this equation are x = 0 or 6 or -2.

I hope that helps.

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

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