02Apolynomials

= Polynomials =

A ** polynomial ** is an expression with
 * only one variable (eg x)
 * one or more terms
 * each term has a non-negative, integer power of the variable

Polynomials are usually named with consecutive capital letters starting at P (ie P, Q, R, etc)

P(x) refers to a polynomial
 * with a name, P
 * with a variable, x


 * Example **

P(x) = 2x 4 + 5x 3 – x 2 + 6x – 7

This is a polynomial where
 * P(x) has 5 terms
 * The highest power of P(x) is 4
 * –7 is a __**constant term**__. It doesn't change when x changes.


 * Remember: **
 * None of the terms may have a power that is negative or a fraction.
 * Only non-negative, integer powers are permitted in a polynomial.


 * The Degree of a Polynomial **

The degree of a polynomial is given by the highest power.

In the example above, P(x) has degree = 4.


 * Polynomial types **


 * A linear polynomial has degree = 1
 * A quadratic polynomial has degree = 2
 * A cubic polynomial has degree = 3
 * A quartic polynomial has degree = 4


 * Sample Polynomials **


 * 10
 * degree = 0 ........ { 10 = 10x 0 }
 * 2x + 5
 * degree = 1 ..... { 2x = 2x 1 }
 * is a __**linear**__ polynomial
 * x 2 – 3x + 5
 * degree = 2
 * is a __**quadratic**__ polynomial
 * 2x 3 + 5x 2 + 3x – 7
 * degree = 3
 * is a __**cubic**__ polynomial
 * x 4 + 5x 3 – 2x + 1
 * degree = 4
 * is a __**quartic**__ polynomial


 * Value of a Polynomial **

The notation P(4) implies substitute x = 4 into P(x) and evaluate the result.


 * Example **

... ... Given the polynomial P(x) = x 2 – 5x ... ... Find the value of P(4)


 * Solution:**

math \\ .\qquad P(4) = (4)^2 - 5(4) \\. \\ . \qquad P(4) = 16 - 20 \\. \\ . \qquad P(4) = -4 math

.