I'm sure every e2 user has, at one point in their life, asked themselves the question, "Is there any way to make
polynomial long division even better than it already is?" Wonder no longer, for here I present to you the
secret to synthetic
division.
Synthetic division is basically long division condensed. It can be used on polynomial equations of any degree as long as the divisor is of the form x + c where c is a constant. For example, if you are given the polynomial expression:
3x^{4} + 2x^{3}  5x^{2}  6x + 4
and you want to divide by
x2. Set up for synthetic division as follows:
2  3 2 5 6 4


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The first number, before the pipe, is c from the divisor. The numbers that follow are the coeffients from the polynomial expression. Note: Every "degree" must be represented, for example if the expression was 3x^{4} + 2x^{3}  6x + 4
, instead of the 5, there should be a zero. Skipping over it entirely is not a
good thing.
Now we shall proceed. First bring the first number down below the line like so.
2  3 2 5 6 4


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3
Now multiply that number by the number in the box, in this case 2. Place that number beneath the next term, add them, and place the result underneath the line.
2  3 2 5 6 4

 6
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3 8
Continue this process, this time multiplying 8 by 2, placing it beneath 5, adding and placing the result beneath the line.
2  3 2 5 6 4

 6 16
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3 8 11
The final result should look like this:
2  3 2 5 6 4

 6 16 22 32
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3 8 11 16 36
How to interpret this strange mess? Easy! The resulting polynomial expressing is characterized by the numbers below the line, which represent the coefficients. In this case, you're left with 3x^{3} + 8x^{2} + 11x + 16, and a remainder of 36.
Additional fun with Synthetic Division:
This is a very useful tool, at least for the Math 101 student, and for anyone needing to factor polynomials with degree greater than 2! Using the Rational Zeros Theorem, you can quickly confirm whether or not a possible zero is a true zero and reduce the expression down to a nice quadratic equation.
Synthetic division is also a fast way to evaluate expressions if you happen to be unable to calculate, say, 18^{5} in your head. The remainder is the value of the expression where x = c.
This is also a fun party trick! Oh wait, no it's not. Unless you go to parties where people factor polynomials for fun, but in that case you should already know this.
As always, /msg me with any suggestions or math errors!