Finding Coefficients of Simultaneous Equations

Date: 07/03/98 at 04:12:59
From: Emma
Subject: Polynomials

The equations ax^4 + bx^3 + c = 0 and cx^4 + bx^3 + a = 0 have a 
common root. Find all possible values of b, if a and c are different 
numbers and a + c = 100.

I am not really sure how to even begin this problem. I have done a 
little work finding factors and factor theorems. Unfortunately, I 
can't seem to see how to apply this and hence solve the question. I 
would appreciate some guidance in coming to the correct answer.
Thanks!

Date: 07/03/98 at 04:53:22
From: Doctor Pete
Subject: Re: Polynomials

Hi,

Here's how to begin.  Suppose the polynomials:

     ax^4 + bx^3 + c
     cx^4 + bx^3 + a

have a common root. Call this root r. Then this gives us two equations:

     ar^4 + br^3 + c = 0
     cr^4 + br^3 + a = 0

Now, r is a constant. You don't know what a, b, and c are, but clearly 
the two equations above form a system. They must both be satisfied by a 
single solution {a, b, c}. Undoubtedly you've done systems of two 
equations in two unknowns, right?  Well, how about 3 equations in 3 
unknowns? Here, we have two equations in 3 unknowns, but actually, 
there is a third:

     a + c = 100

This gives us enough information to solve for b. We will suppose a, b, 
and c are real numbers. Then if we take the first two equations and 
subtract the second from the first, we obtain:

     (a-c)r^4 + 0 + (c-a) = 0

Since a is not equal to c, we may rearrange and divide by a-c, giving

     r^4 = 1

Substituting back into, say, the first equation, we obtain:

     a(1) + br^3 + c = (a+c) + br^3 = 100 + br^3 = 0

or b = -100/r^3.

If r^4 = 1, and b is real, then what are the possible values of r^3?  
(There are two.) Hence, what are the possible values of b?

Notice that the process of solving the problem started with what you 
knew about the problem. The next step was giving a name to other 
quantities or properties mentioned in the question, for example, saying 
r is the common root of the two equations. Third, apply as many 
properties or relationships you can - this is where I wrote down the 
two equations equalling 0 when x = r. Next, try to simplify or solve, 
using any additional information (a+c = 100, for instance). The key was 
to notice that this was a question about solving simultaneous equations 
in more than one unknown, not necessarily about factoring.  

The reason I knew this was that I came across 3 equations (the two 
polynomials, and the a + c condition), and 3 unknowns: a, b, c.  
Furthermore, it's impossible to factor the given polynomials when 
you don't even know their coefficients (which, after all, was what you 
were trying to find). Another critical part was knowing how to solve 
the equations, obtaining a condition on the value of r, and using this 
to obtain the possible values of b. In this case, even though I knew r 
was a constant, I didn't know what r could be, which told me that in 
order to find b I needed to find r first.

As you begin to learn more theorems and techniques in math, it becomes 
more difficult to concentrate on problem-solving methods, rather than 
simply applying the theorems blindly. But with more practice, you'll 
get the hang of it - the more you work with the ideas, the better 
you'll get at intuitively sensing which direction you need to go in, 
how to start, and what tools you'll need to solve the problem. Even 
after you solve a problem, it helps to think about the process, and ask 
yourself questions. For instance, can you answer why a and c are not 
uniquely determined, even though you can find the possible values of b?  
Also, what if a = c?  Is there any way of determining b in this case?

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