polynomial.nbPolynomials are functions of the form:
f[x] := a[0]*x^0 + a[1]*x^1 + a[2]*x^2 + ... + a[n]*x^n
Or in summation notation:
f[x] := Sum[a[n]*x^n, {0,1,n}]
where a[_] are real numbers and n is a positive integer. For example, the following are polynomials.
Each element in the addition is called a term. For example, in -2*x^5 -4*x^3, there are two terms: -2*x^5 and -4*x^3. The constant in each term is called the coefficient of that term. For example, in the term -2*x^5, -2 is the coefficient. Each term is associated with a integer called the power or degree of that term. For example, in -2*x^5, the degree is 5. A polynomial is associated with a integer called the power or degree of the polynomial, which is defined to be the the max of the degree of all its terms. For example, the degree of the polynomial -2*x^5 -4x^3 is 5.
The concept of degree of a polynomial is important, because it gives us info about the behavior of the polynomial on the whole. We'll see it when we study the graph of polynomial.
A polynomial can be plotted as to show its properties visually. A polynomial is plotted as the curve y==f[x]
polynomial_t2.gcf
The concept of polynomial functions goes way back to perhaps Babylonians times, since for example as simple a need of computing the area of a square y==x^2 is a polynomial, and is needed in buildings and survey, fundamental to core civilization. The Pythagorean theorem x^2+y^2==z^2 is also a polynomial equation, and much basic number theory have been expressed algorithmetically in Greek or pre-Greek era.
The modern concept of polynomial as a function of integer powers and their symbolic manipulation is developed in 1600s and 1700s. Finding solutions of polynomials as ready-made formulas is a spetacular chapter in the history of mathematics, culminating in the birth of Complex Numbers and Group Theory.
f[x] := Sum[a[n]*x^n, {0,1,n}]
The graph of a polynomial b*x^n looks like a Parabola if n is even. Otherwise, it is s-shaped. These facts can be easily perceived if we look at the behavior of x^n for large values of n. If n is even, x^n is always positive even if x is negative. When n is odd, x^n is negative when x is negative. The coefficient b scales the curve vertically, and if b is negative, it flips the curve along the y-axis.
Given a polynomial f[x] := Sum[a[n]*x^n, {0,1,n}] the overall behavior is its highest term a[n]*x^n...
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See: Websites on Plane Curves, Printed References On Plane Curves.
Wikipedia: polynomial↗.
© 2004-2008 by Xah Lee.
