Chapter5

# Chapter5 - Pre-Calculus Chapter 5A Chapter 5A Polynomial...

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© : Pre - Calculus - Chapter 5A Chapter 5A - Polynomial Functions Definition of Polynomial Functions A polynomial function is any function p x of the form p x p n x n p n 1 x n 1 p 2 x 2 p 1 x p 0 , where all of the exponents are non-negative integers. The coefficients of the various powers of x ,thatis ,the p i ’s are assumed to be known real numbers, and p n 0. The degree of this polynomial function is n .If n is even the polynomial is said to be of even degree and if n is odd, the polynomial is said to be of odd degree. The coefficient p 0 is called the constant term and the non-zero p n is called the leading coefficient . By a polynomial of degree 0, we mean a non-zero constant function. The zero polynomial function is sometimes said to have degree equal to − . In the table below we list a few polynomials and their degrees. polynomial degree constant term leading coefficient p x 50 5 5 p x 3 x 2 82 83 p x 6 x 5 x 4 5 x 95 9 6 p x 2 x 126 126 0 2 We have studied two examples of polynomial functions already. Linear functions ( f x mx b , m 0) are polynomial functions of degree 1, and quadratic functions ( f x ax 2 bx c , a 0) are polynomial functions of degree 2. Below are some examples of polynomials of degree 3, 4, and 5. Example 1 :L e t p x 5 6 x 13 x 3 . What is the degree of p x ? What are the leading coefficient and constant terms? Solution: The degree of p x 5 6 x 13 x 3 is 3. The largest exponent. The leading coefficient is 13 and the constant term is 5. Example 2 e t p x 3 x 4 3 x 3 5 x . What is the degree of p x ? What are the leading coefficient and constant terms? Solution: The degree of p x 3 x 4 3 x 3 5 x is 4. The leading coefficient is 3, and the constant term, the coefficient of x 0 ,is0 . Example 3 :I f f x 16 3 x 2 27 x 4 5 x 7 8 x 13 , what are the degree, leading coefficient, and constant term equal to? Solution: The degree of f x 16 3 x 2 27 x 4 5 x 7 8 x 13 is 13. The leading coefficient is 8, and the constant term is 16. © : Pre - Calculus

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© : Pre - Calculus - Chapter 5A Behavior of Polynomial Functions In this page we will discuss how the values p x of polynomial functions behave for large values of x , both positive and negative. It turns out that the behavior of polynomial functions for large values of x is determined by the degree of the polynomial. Polynomials of odd degree behave one way, think of the graph of a linear function, and polynomials of even degree behave in a different way, picture the graph of a quadratic function.
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Chapter5 - Pre-Calculus Chapter 5A Chapter 5A Polynomial...

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