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Mathematics LibreTexts

8.3: Optional section- Synthetic division

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When dividing a polynomial f(x) by g(x)=xc, the actual calculation of the long division has a lot of unnecessary repetitions, and we may want to reduce this redundancy as much as possible. In fact, we can extract the essential part of the long division, the result of which is called synthetic division.

Example 8.3.1

Our first example is the long division of 5x3+7x2+x+4x+2.

Solution

clipboard_e0429e2e38938c4948dc456cb9813bfd6.png

Here, the first term 5x2 of the quotient is just copied from the first term of the dividend. We record this together with the coefficients of the dividend 5x3+7x2+x+4 and of the divisor x+2=x(2) as follows:

5714 (dividend)(5x3+7x2+x+4)2 (divisor)(x(2))5 (quotient)

The first actual calculation is performed when multiplying the 5x2 term with 2, and subtracting it from 7x2. We record this as follows.

clipboard_e12d78d348a9a2200177b7d7762e872ee.png

Similarly, we obtain the next step by multiplying the 2x by (3) and subtracting it from 1x. Therefore, we get:

clipboard_eab5c90d60c6540c61329459e499468af.png

The last step multiplies 7 times 2 and subtracts this from 4. In short, we write:

clipboard_ee43fd018237b1db279ed1fa299792eed.png

The answer can be determined from these coefficients. The quotient is 5x23x+7, and the remainder is 10.

Example 8.3.2

Find the following quotients via synthetic division.

  1. 4x37x2+4x8x4
  2. x4x2+5x+3

Solution

  1. We need to perform the synthetic division. 4748416361604940152

    Therefore we have

    4x37x2+4x8x4=4x2+9x+40+152x4

  2. Similarly, we calculate part (b). Note that some of the coefficients are now zero. 1010533924721382477

    We obtain the following result:

    x4x2+5x+3=x33x2+8x24+77x+5

Note

Synthetic division only works when dividing by a polynomial of the form xc. Do not attempt to use this method to divide by other forms like x2+2.


This page titled 8.3: Optional section- Synthetic division is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

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