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Polynomials Basics.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \begin{document} {\large Polynomials} \begin{align*} \text{\bf The Basics:}\quad& P(x)=\sum_{r=0}^n a_k x^k=a_n x^n+a_{n-1}x^{n-1}+\ldots+a_1 x+a_0\:,\quad \text{where $a_0, a_1, \ldots, a_n$ are constants and $a_n\neq 0$.}\\ % \text{Terminology:}\quad &a_n x^n\quad\ldots\text{ leading term}\\ &a_n\quad\ldots\text{ leading coefficient}\\ &n\quad\ldots\text{ degree}\qquad\text{i.e. }\deg P(x)=n\\ &\text{A polynomial of degree zero ($\deg P(x)=0$) is called the \it zero polynomial\rm: }Z(x)=0\:.\\ &\text{A polynomial of degree one ($\deg P(x)=1$) is called a \it constant polynomial\rm: }Z(x)=a_0\neq 0\:.\\ &\text{A polynomial of degree two ($\deg P(x)=2$) is called a \it linear polynomial\rm: }Z(x)=a_1 x +a_0\:,\quad a_1\neq 0\:.\\ &\qquad\text{(Its graph is a staight line, hence the word \it linear\rm.)}\\ &\text{A polynomial of degree three ($\deg P(x)=3$) is called a \it quadratic polynomial\rm: }Z(x)=a_2 x^2+a_1 x +a_0\:,\quad a_2\neq 0\:.\\ &\qquad\text{(Its graph is a parabola.)}\\ % &P(x)\text{ is a \it monic polynomial \rm if the leading coefficient }a_n=1\:.\\ &\qquad\text{Note: A monic polynomial represents a class of polynomials with their coefficient in ratio.}\\ &\qquad\text{e.g. }8x^3+4x^2-2x+6\quad\text{is in the same class as}\quad 4x^3+2x^2-x+3\:,\\ &\qquad\text{as the monic polynomial for both are}\quad x^3+\tfrac{1}{2}x^2-\tfrac{1}{4}x+\tfrac{3}{4}\:.\\ \\ \text{Identically }&\text{Equal Polynomials:}\\ \end{align*} \end{document}