An Effective Alternative to Synthetic Division
- Introduction
- Welcome!
- http://bit.ly/MoreDivisionPlease - (this presentation site)
- Motivation
- am I multiplying or dividing?
- that which remains
- consistency
- Rational Functions (Algebra)
- quadratic divisor
- Power Series (Calculus)
- quadratic divisor
- multiply and divide power series
- Generating Functions (Discrete Mathematics)
- generalized combinations
- unordered counting
- ordered counting
- Supporting materials
- Presentation Examples.pdf - calculations of examples
- Desmos illustrations
- An Effective Alternative to Synthetic Division.nb - Mathematica notebook (printed)
Introduction
When it comes to synthetic division, are you an "adder" or a "subtractor"? Come learn to be a "filler"! Have you given up on polynomial division, synthetic or otherwise? Come be reenergized!
Learn an effective method for dividing any two polynomials, even any two power series, that can produce either the quotient and remainder or the power series representation. Applications to algebra, calculus, and discrete mathematics will be demonstrated.
mathematics > presentations > an effective alternative to synthetic division > motivationMotivation
Using tables of coefficients to multiply or divide polynomials.
\( \begin{eqnarray*} x^{2}+5x+6=(x+2)(x+3)\\ \end{eqnarray*} \)
\( \begin{eqnarray*} \frac{x^{2}+5x+6}{x+2} & = & x+3\\ \end{eqnarray*} \)
\( \begin{eqnarray*} \frac{x^{2}+5x+7}{x+2} & = & x+3+\frac{1}{x+2}\\ & = & 3+x+\frac{1}{2}\left(\frac{1}{1-(-x/2)}\right)\\ & = & 3+x+\frac{1}{2}\left(1-\frac{x}{2}+\frac{x^{2}}{4}-\frac{x^{3}}{8}+\cdots\right)\\ \end{eqnarray*} \)
\( \begin{eqnarray*} \frac{x^{2}+5x+7}{x+2} & = & \frac{7}{2}+\frac{3}{4}x+\frac{1}{8}x^{2}-\frac{1}{16}x^{3}+\frac{1}{32}x^{4}+\cdots\\ \end{eqnarray*} \)
(calculation | Desmos)
mathematics > presentations > an effective alternative to synthetic division > rational functionsRational Functions
Graph the following rational function
\( \begin{eqnarray*} \frac{x^4-3x^{3}-x^{2}+5x+3}{2x^{2}-x-2} & = & \frac{1}{2}x^{2}-\frac{5}{4}x-\frac{5}{8}+\frac{\frac{15}{8}x+\frac{7}{4}}{2x^{2}-x-2}\\ \end{eqnarray*} \)
(calculation | Desmos)
mathematics > presentations > an effective alternative to synthetic division > power seriesPower Series
Approximate with a Maclaurin Series
\( \begin{eqnarray*} \frac{x^3-3x^{2}+x-2}{x^{2}-2x+4} & = & -\frac{1}{2}-\frac{5}{8}x^2-\frac{1}{16}x^{3}+\frac{1}{8}x^{4}+\cdots \\ \end{eqnarray*} \)
Multiply and Divide Power Series
\( \begin{eqnarray*} e^{-2x}\sin 3x & = & \left(1-2x+2x^{2}+\cdots\right)\left(3x-\frac{9}{2}x^{3}+\frac{81}{40}x^{5}+\cdots\right) \\ & = & 3x-6x^{2}+\frac{3}{2}x^{3}+5x^{4}+\cdots \\ & & \\ \frac{\sin 3x}{e^{2x}} & = & \frac{3x-\frac{9}{2}x^{3}+\frac{81}{40}x^{5}+\cdots}{1-2x+2x^{2}+\cdots} \\ & = & 3x-6x^{2}+\frac{3}{2}x^{3}+5x^{4}+\cdots \\ \end{eqnarray*} \)
(calculation | Desmos)
mathematics > presentations > an effective alternative to synthetic division > generating functionsGenerating Functions
Unordered Combination
\( \begin{eqnarray*} (1 + x + x^{2} + \cdots) (1 + x + x^{2} + \cdots) (1 + x + x^{2} + \cdots) & = & (1+2x+3x^{2}+4x^{3}+5x^{4}+\cdots)(1 + x + x^{2} + x^{3} + x^{4} +\cdots) \\ & = & 1+3x+6x^{2}+10x^{3}+15x^{4}+21x^{5}+\cdots\\ \end{eqnarray*} \)
Another Unordered Combination
\( \begin{eqnarray*} (1 + x + x^{2} + \cdots) (1 + x^{2} + x^{4} + \cdots)(1 + x^{5} + x^{10} + \cdots) & = & (1+x+2x^{2}+2x^{3}+\cdots)(1 + x^{5} + x^{10} + \cdots)\\ & = & 1+x+2x^{2}+2x^{3}+3x^{4}+4x^{5}+\cdots\\ \end{eqnarray*} \)
Ordered Combination
\( \begin{eqnarray*} 1 + (x + x^{2} + x^{5}) + (x + x^{2} + x^{5})^{2} + (x + x^{2} + x^{5})^{3} + \cdots & = & \frac{1}{1-x-x^{2}-x^{5}}\\ & = & 1+x+2x^{2}+3x^{3}+5x^{4}+\cdots\\ \end{eqnarray*} \)