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Synthetic Division Examples (page 2 of 4)

Sections: Introduction, Worked examples, Finding zeroes, Factoring polynomials


  • Complete the indicated division.
    • -2  |__2_-3_-5__3__8_

    For this first exercise, I will display the entire synthetic-division process step-by-step.

    First, carry down the "2" that indicates the leading coefficient:

       

     Carry down the &quo

      

    Multiply by the number on the left, and carry the result into the next column:

       

     Multiply (2)(2) to get 4 in the next column.

      

     
    Add down the column:

      

     Add down: -3 + 4 = 1.

      

    Multiply by the number on the left, and carry the result into the next column:

       

     Multiply: (2)(1) = 2; carry this into the next column.

      

     
    Add down the column:

       

     Add down:  -5 + 2 = -3.

      

    Multiply by the number on the left, and carry the result into the next column:

       

     Multiply: (2)(-3) = -6; carry this into the next column.

      

     
    Add down the column:

       

     Add down:  3 + (-6) = -3.

      

    Multiply by the number on the left, and carry the result into the next column:

       

     Multiply: (2)(-3) = -6; carry this into the next column.

      

     
    Add down the column for the remainder:

       

     Add down:  8 + (-6) = 2, a non-zero remainder.

      

     
    The completed division is:

       

     Result: 2  1 -3 -3  2, or 2x^3 + x - 3x - 3, remainder 2.

This exercise never said anything about polynomials, factors, or zeroes, but this division says that, if you divide 2x4 3x3 5x2 + 3x + 8 by x 2, then the remainder will be 2, and therefore x 2 is not a factor of 2x4 3x3 5x2 + 3x + 8, and x = 2 is not a zero (that is, a root or x-intercept) of the initial polynomial. Copyright Elizabeth Stapel 2002-2011 All Rights Reserved

  • Divide 3x3 2x2 + 3x 4 by x 3 using synthetic division.
    Write the answer in the form "
    q(x) + r(x)/d(x) ".
  • This question is asking me, in effect, to convert an "improper" polynomial "fraction" into a polynomial "mixed number". That is, I am being asked to do something similar to converting the improper fraction 17/5 to the mixed number 3 2/5, which is really the shorthand for the addition expression "3 + 2/5".

    To convert the polynomial division into the required "mixed number" format, I have to do the division; I will show most of the steps.

    First, write down all the coefficients, and put the zero from x 3 = 0 (so x = 3) at the left.
       

      

     
    Set up the synthetic division
     

     
    Next, carry down the leading coefficient:

      

    Carry down the 3

        

    Multiply by the potential zero, carry up to the next column, and add down:

      

    bottom row: 3  7

       

     
    Repeat this process:

      

    bottom row:  3  7  24

       

     
    Repeat this process again:

      

    bottom row:  3  7  24  68

    As you can see, the remainder is 68. Since I started with a polynomial of degree 3 and then divided by x 3 (that is, by a polynomial of degree 1), I am left with a polynomial of degree 2. Then the bottom line represents the polynomial 3x2 + 7x + 24 with a remainder of 68. Putting this result into the required "mixed number" format, I get the answer as being:

      3x  + 7x + 24 + 68/(x - 3)

It is always true that, when you use synthetic division, your answer (in the bottom row) will be of degree one less than what you'd started with, because you have divided out a linear factor. That was how I knew that my answer, denoted by the "3  7  24" in the bottom row, stood for a quadratic, since I had started with a cubic.

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Cite this article as:

Stapel, Elizabeth. "Synthetic Division Examples." Purplemath. Available from
    http://www.purplemath.com/modules/synthdiv2.htm. Accessed
 

 



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