Question 242898
{{{(3x^3+4x-1) / (x^2+1)}}}
<pre><font size = 4 color = "indigo"><b>
You have to put in placeholder zeros for the missing powers in both the 
divisor and the dividend, and deal with 0 placeholders all through
the long division process.

You have to write {{{3x^3+4x-1}}} as {{{3x^2+0x^2+4x-1}}} and
you have to write {{{x^2+1}}} as {{{x^2+0x+1}}}

Then you write this:

         
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1 

Divide {{{(3x^3)/x^2}}} getting {{{3x}}} and write this
above the line in line with the 4x:

                        3x
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1 

Now multiply 3x by x² + 0x + 1, getting 3x³ + 0x² + 3x,
and write it below like this, keeping like powers of x
lined up, then draw a line underneath.

                        3x
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1
            3x³ + 0x² + 3x
            --------------
                   
Now subtract (3x³ + 0x² + 4x) - (3x³ + 0x² + 3x) = 0x² + x.

(Not you must keep the placeholder zero for the x² term:


                        3x
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1
            3x³ + 0x² + 3x
            --------------
                  0x² +  x

Now bring down the next (last) term -1:

                        3x
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1
            3x³ + 0x² + 3x
            --------------
                  0x² +  x - 1

Next divide {{{0x^2/x^2}}} getting 0, so write + 0
on top above the -1:

                        3x + 0
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1
            3x³ + 0x² + 3x
            --------------
                  0x² +  x - 1

Multiply 0 by x² + 0x + 1 getting 0x² + 0x + 0 and
write it at the bottom. Then draw a line:

                        3x + 0
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1
            3x³ + 0x² + 3x
            --------------
                  0x² +  x - 1
                  0x² + 0x + 0
                  ------------

Subtract: (0x² +  x - 1) - (0x² + 0x + 0) = x - 1, so

write that at the bottom:

                        3x + 0
           ------------------- 
x² + 0x + 1)3x³ + 0x² + 4x - 1
            3x³ + 0x² + 3x
            --------------
                  0x² +  x - 1
                  0x² + 0x + 0
                  ------------
                         x - 1

Now the final answer is gotten by adding the fraction
{{{(remainder)/(divisor)}}} to the quotient:

{{{3x+0+(x-1)/(x^2+0x+1)}}}

Now we can drop the place holder zeros and get:

{{{3x+(x-1)/(x^2+1)}}}

Edwin</pre>