Question 141539
 Use synthetic division to perform the division 
(2x^4 - 3x^3 + 2x - 5) divided by (3x + 2) 
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We have to put in a placeholder of {{{0x^2}}}

{{{(2x^4 - 3x^3 + 0x^2 + 2x - 5)}}}÷{{{(3x + 2)}}}

Since synthetic division can only be used to divide 
by expressions in the form x - A, we have to make two
divisions:

1. First we factor 3 out of {{{3x+2}}} and get {{{3(x+2/3)}}}
2. So we divide first synthetically by {{{(x+2/3)}}} 
3. Then  we divide by 3.

Dividing first by {{{x+2/3}}}

-2/3| 2    -3     0      2      -5
    |<u>     -4/3  26/9  -52/27   -4/81 </u>   
      2  -13/3  26/9    2/27  -409/81


That gives 

      {{{2x^3 - (13/3)x^2 + (26/9)x + (2/27) +((-409)/81)/(x+2/3)}}}

First we'll simplify that fraction on the end:

Multiply top and bottom by 81:

{{{(81((-409)/81))/(81(x+2/3))=(-409)/(81x+54)}}}

So dividing by {{{x+2/3}}} gives:

      {{{2x^3 - (13/3)x^2 + (26/9)x + (2/27) + (-409)/(81x+54)}}}

Now we must divide every term by 3, which is the same
as multiplying by {{{1/3}}}:

      {{{(2/3)x^3 - (13/9)x^2 + (26/27)x + (2/81) -409/(3(81x+54))}}}

What a terrible problem!  Are you sure you didn't copy something
wrong?

Edwin</pre>