Question 93176
I need help understanding the polynomials division problems
with exponents. 

14y+8y˛+ył+12 ÷ 6+y 
<pre><font size = 5><b>
First rearrange the terms of both so that the 
powers of y are descending and the constant is 
on the far right:

ył + 8y˛ + 14y + 12 ÷ y + 6

Write this:
                                     
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
              
Divide the y on the far left into the ył,
getting y˛, so write that on top right above
the y˛ in the term 8y˛:
          
            y˛             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
    
Now multiply that y˛ by both the y and the 6,
getting ył + 6y˛ and write than inder the ył + 8y  
write that underneath and 


            y˛             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
         
Now subtract the 8y˛ MINUS 6y˛ gives 2y˛, so write
that under the line.  (When you subrtact the ył MINUS
ył you just get 0, so you don't write anything under
that:


            y˛             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ 
         
Now you bring down the + 14y˛


            y˛             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           
Now divide the y on the far left into the
2y˛ at the bottom, getting 2y, which you write
as + 2y above the top line above the + 14y

            y˛ +  2y             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
       
Next you multiply the 2y by the y + 6, getting
2y˛ + 12y.  Write that at the bottom and draw
a line under it:


            y˛ +  2y             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                
Subtract 14y MINUS 12y getting 2y, write that
under the line.  Don't write anything under
the 2y˛ MINUS 2y˛ because that's just 0.

            y˛ +  2y             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y 
                 
                  
Bring down the + 12

                        
            y˛ +  2y             
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y + 12

Divide the y on the far left into the 2y at 
the bottom, getting 2.  Write + 2 above the
top line, above the + 12


            y˛ +  2y +  2            
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y + 12

Multiply the 2 by the y + 6, getting 2y + 12.  Write
that at the bottom and underline it:

              
            y˛ +  2y +  2            
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y + 12
                  2y + 12
                  ———————
                        
When you subtract you get 0.  So you can
write 0 at the bottom if you like:

            y˛ +  2y +  2            
      ———————————————————
y + 6)ył + 8y˛ + 14y + 12
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y + 12
                  2y + 12
                  ———————
                        0

The remainder is 0, so the quotient (answer)

is     y˛ + 2y + 2

It is important to keep like powers of x vertically
straight. 

Notice that the <font color = "red">ył</font>'s are 
lined up straight vertically:

            y˛ +  2y +  2            
      ———————————————————
y + 6)<font color = "red">ył</font> + 8y˛ + 14y + 12
      <font color = "red">ył</font> + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y + 12
                  2y + 12
                  ———————
                        0

Notice that the <font color = "red">y˛</font>'s are 
lined up straight vertically:

            <font color = "red">y˛</font> +  2y +  2            
      ———————————————————
y + 6)ył + 8<font color = "red">y˛</font> + 14y + 12
      ył + 6<font color = "red">y˛</font>
      ————————
           2<font color = "red">y˛</font> + 14y
           2<font color = "red">y˛</font> + 12y
           —————————
                  2y + 12
                  2y + 12
                  ———————
                        0

Notice that the y's are lined up vertically:

            y˛ +  2<font color = "red">y</font> +  2            
      ———————————————————
y + 6)ył + 8y˛ + 14<font color = "red">y</font> + 12
      ył + 6y˛
      ————————
           2y˛ + 14<font color = "red">y</font>
           2y˛ + 12<font color = "red">y</font>
           —————————
                  2<font color = "red">y</font> + 12
                  2<font color = "red">y</font> + 12
                  ———————
                        0

And the constant numbers are also lined up
vertically.

            y˛ +  2y +  <font color = "red">2</font>            
      ———————————————————
y + 6)ył + 8y˛ + 14y + <font color = "red">12</font>
      ył + 6y˛
      ————————
           2y˛ + 14y
           2y˛ + 12y
           —————————
                  2y + <font color = "red">12</font>
                  2y + <font color = "red">12</font>
                  ———————
                        <font color = "red">0</font>


I point this out because it is important to keep 
like terms lined up vertically when dividing 
polynomials.

Edwin</pre>