8.3: Optional section- Synthetic division
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When dividing a polynomial f(x) by g(x)=x−c, 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.
Our first example is the long division of 5x3+7x2+x+4x+2.
Solution
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.
Similarly, we obtain the next step by multiplying the 2x by (−3) and subtracting it from 1x. Therefore, we get:
The last step multiplies 7 times 2 and subtracts this from 4. In short, we write:
The answer can be determined from these coefficients. The quotient is 5x2−3x+7, and the remainder is −10.
Find the following quotients via synthetic division.
- 4x3−7x2+4x−8x−4
- x4−x2+5x+3
Solution
- We need to perform the synthetic division. 4−74−8416361604940152
Therefore we have
4x3−7x2+4x−8x−4=4x2+9x+40+152x−4
-
Similarly, we calculate part (b). Note that some of the coefficients are now zero. 10−105−3−39−24721−38−2477
We obtain the following result:
x4−x2+5x+3=x3−3x2+8x−24+77x+5
Synthetic division only works when dividing by a polynomial of the form x−c. Do not attempt to use this method to divide by other forms like x2+2.