Sunday Comix (Synthetic Division)
2001-09-04 - 1:36 p.m.
Sorry I didn't do Sunday Comix last week, it completely slipped my mind.
In third grade, I was really freaking smart. But, since I was living in Maine, they said I was a �smat� kid instead. Technicality. Anyways, in the middle of third grade, for whatever reason, I forgot how division works. I could still divide simple integers like 306/4, but did so more out of habit than actually thinking the problem out. It was horrible, and I never told any of my teachers...there was a short downward spike in my grades while I relearned longhand division. It was odd...as if I had learned shorthand writing, and used it for so long that I had forgotten my alphabet. So I taught myself some math, and found that I sorta liked learning math on my own. I vowed never again to ask a question in a math class that I didn�t already know the answer to, and I�ve never really regretted that decision. By fourth grade, I was put into a �gifted and talented� class with one other kid in my school. Basically, we got to do semi-telepathic exercises, refine our writing, play with the chemical properties of kool-aid, and make stuff out of tubes of glass. The other person in the programe, Emily, was one of Sarah�s friends, and soon became one of mine. At the end of sixth grade, I moved to Pittsfield, thinking that I�d never see her again...but the world turns at mysterious angles, and it just so happened that she graduated from the same magnet school as me (a year earlier). But I never really spent much time with her, in the one year that we were going to high school together, for whatever reason...That said, let�s jump into some mathematics; specifically, synthetic division (which is no longer required learning in high school algebra).
Synthetic division is an arithmetic system used for polynomials with simple roots such as the monomial (x - h). The system works because a polynomial can theoretically be divided by a monomial and the result will be the remaining roots. Synthetic division is most commonly used when the result of the division will lead to a polynomial that can be used in the quadratic equation. Also, if one is trying to find f(x) = A*x^3+B*x^2+C*x+D, synthetic division provides an efficient way of calculating f(a). The process of synthetic division can also be proved through a simplification of nested multiplication [f(x) = x^3-5x^2+2x-10 is equivalent to f(x) = x(x(x-5)+2)-10].
First, you have to find a root of the polynomial. In the example x^3 - 3x^2 + x + 1 = 0, an obvious root is 1. Therefore, the monomial (x - 1) can be divided from the polynomial. Put the root aside and within view, and write the coefficients of each power of x in the sequence in which they appear (if a power of x is not present, make sure to put in a zero). In the examples below, x^3 - 5x^2 + 12 = 0 (with the assumed root of three) is used...
1 -5 0 12 | X=3
Then draw a line and bring down the first coefficient...
1 -5 0 12 | X=3
____________________
1
After that, you multiply by the root and add the product to the next coefficient in the top line...
1 -5 0 12 | X=3
0 3 -6 -18
____________________
1 -2 -6 -6
The -6 is called the remainder of the division. If an assumed root resulted in a zero, then the assumed root (h) is an actual root, and the polynomial may be rewritten as (x - h)*(coefficients of x under the line). For example, if we used the root of two, the polynomial could be expressed as (x - 2)(1x^2 - 1x -4). The quadratic equation can be used from here to find the two remaining roots.
Synthetic division is also a simplified version of polynomial long division (because of the simple monomial being divided), with the following changes: erase the Xs, the lower duplicates when a number is subtracted from itself, and move the first product of division into the solution. I know, you have no idea what polynomial division is...yer just gonna have to learn some things on yer own. Today�s broadcast was brought to you by the number e, the imaginary letter i, and the question y. (Next time I do a math-orientated Sunday Comix, I hope to cover Fundamental Calculus...and then maybe we can talk about non-Euclidean geometries.)
