Question 81556
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Solution by Edwin McCravy:

I don't think Dr. Who used the method you were taught, 
so I thought I'd do it the way I think you were taught.
Also, he didn't tell you how to check it.
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I have the following: {{{(2x^2-18x+8)/(2x-4)}}}.  The instructions are to divide.  
I believe the answer to be x-7 with a remainder of 10/x-2.  However, I am
not sure how to check if this answer is correct. Would you please help me 
solve?

Thank you. 
<pre><font size = 5><b>
So you started with  

 2x² - 18x + 8
---------------  
    2x - 4

and got

          10
x - 7 + -------
         x - 2 

The best way to check to see if that is right
is to:
 
1. substitute 0 for x in the original.
2. substitute 0 for x in the final answer.
3. If you don't get the same thing then you
   have made an error.
4. If you got the same thing in steps 1 and 2,
   then you still may have made an error, so:
5. Substitute 1 for x in the original. 
6. Substitute 1 for x in the final.
7. If you get the same thing then you are 99%
   certain you are correct. If not, you have made 
   an error.
(Note: if the divisor is 0 when you substitute 1 in
step 5, then substitute 2 instead of 1 in 6 and 7.

Let's check yours:

1. substitute 0 for x in the original.

 2x² - 18x + 8
---------------  
    2x - 4

 2(0)² - 18(0) + 8       8 
------------------- =  ---- = -2 
    2(0) - 4            -4

2. substitute 0 for x in your final answer.

          10
x - 7 + -------
         x - 2 

          10
0 - 7 + ------- = -7 - 5 = -12
         0 - 2 

Oh oh!  Those are not the same.  So we don't
need to substitute 1.  You made an error.

Let's do it again:

       <u>        x -  7</u> 
2x - 4)2x² - 18x +  8
       <u>2x² -  4x</u>
            -14x +  8
            <u>-14x + 28</u>
                  -20

Write the -20 over the divisor and add to the
quotient.  The answer, before simplifying, is

          -20
x - 7 + ---------
         2x - 4 

On the bottom you can factor out a 2

           -20
x - 7 + ----------
         2(x - 2) 

Now you can cancel the 2 on the bottom into
the -20 on top and get -10 

           -10 
           <s>-20</s>
x - 7 + ----------
         <s>2</s>(x - 2) 
         1

          -10
x - 7 + --------
         x - 2

So it looks like you made an error in the sign of
the remainder.  You can now bring the negative sign
out front of the fraction and have:

           10
x - 7 - --------
         x - 2

as our new final answer.

Now let's check it:

1. substitute 0 for x in the original.

 2x² - 18x + 8
---------------  
    2x - 4

 2(0)² - 18(0) + 8       8 
------------------- =  ---- = -2 
    2(0) - 4            -4

2. substitute 0 for x in our new final answer.

          10
x - 7 - -------
         x - 2 

          10
0 - 7 - ------- = -7 - (-5) = -7 + 5 = -2
         0 - 2 


3. We got the same thing.
4. Since we got the same thing in steps 1 and 2,
   then:
5. Substitute 1 for x in the original. 

 2x² - 18x + 8
---------------  
    2x - 4

 2(1)² - 18(1) + 8      -8 
------------------- =  ---- = +4 
    2(1) - 4            -2


6. Substitute 1 for x in our new final answer:

          10
x - 7 - -------
         x - 2 

          10            10
1 - 7 - ------- = -6 - ---- = -6 - (-10) = -6 + 10 = +4  
         1 - 2          -1

7. We get the same thing so we are now 99% sure that
   this is correct.  This is not an absolutely perfect
   check, but the first three steps (substituting 0)
   will nearly always catch an error.
   
Edwin</pre>