Algebra Refresher

Date: 12/31/97 at 11:49:42
From: Aileene Cottrell
Subject: Need Help Immediately

Dear Dr. Math:

I am trying to get into an algebra class in college and am refreshing 
my memory of what I learned in high school. Currently, I am trying to 
figure out how to solve this equation: (x^3 + 3x^2 - 4x + 3) / (x+1).  
If you can walk me through this and give me a formula that I can 
always apply to get to the right answer, I'd really appreciate it.

Thanks,
Aileene

Date: 12/31/97 at 12:11:47
From: Doctor Rob
Subject: Re: Need Help Immediately

First of all, this is not an equation. There is no "=" sign. If you
mean to simplify this, it is already in simplest form. If you mean to
divide, giving quotient and remainder, here is how to proceed, using 
long division:

         -----------------------
   x + 1 ) x^3 + 3*x^2 - 4*x + 3

Divide the leading term "x" of the divisor into the leading term "x^3" 
of the dividend. The quotient is x^2. Put that above the 3*x^2 term of 
the dividend, and above the line. Multiply that x^2 times the divisor,
putting the result below the dividend, and subtract. Bring down the 
next term from the dividend into the current remainder. Your work 
should look like this:

                   x^2
         -----------------------
   x + 1 ) x^3 + 3*x^2 - 4*x + 3
           x^3 +   x^2
          -------------
                 2*x^2 - 4*x

Now divide x into 2*x^2 to get the next term of the quotient. It is 
2*x. Put it above the -4*x term of the dividend, and above the line.  
Multiply that 2*x times the divisor, putting the result below the 
current remainder, and subtract. Bring down the next term from the 
dividend into the current remainder. Now you should have:

                   x^2 + 2*x
         -----------------------
   x + 1 ) x^3 + 3*x^2 - 4*x + 3
           x^3 +   x^2
          -------------
                 2*x^2 - 4*x
                 2*x^2 + 2*x
                -------------
                       - 6*x + 3

Now divide x into -6*x to get the next term of the quotient. It is -6.
Put it above the +3 term of the dividend, and above the line. Multiply 
that -6 times the divisor, putting the result below the current 
remainder, and subtract.  Now you should have:

                   x^2 + 2*x - 6
         -----------------------
   x + 1 ) x^3 + 3*x^2 - 4*x + 3
           x^3 +   x^2
          -------------
                 2*x^2 - 4*x
                 2*x^2 + 2*x
                -------------
                       - 6*x + 3
                       - 6*x - 6
                      -----------
                               9

You are done, because 9 involves no x's at all, and the divisor x + 1
has x to the first power, so you cannot divide x + 1 into 9.  The
quotient is the top line, x^2 + 2*x - 6, and the remainder is the 
bottom line, 9.  This means (x^3 + 3*x^2 - 4*x + 3)/(x + 1) is x^2 + 
2*x - 6 with remainder 9, or, in other words,

   (x^3 + 3*x^2 - 4*x + 3)/(x + 1) = x^2 + 2*x - 6 + [9/(x+1)].

Notice that, reading down the columns, every term has the same 
exponent of x.  This is the best way to arrange your work. Keeping 
your columns straight will help you keep from getting confused.

I hope that this was what you wanted. If not, write back to us again.

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