Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s are constants, a n 0; n is a nonnegative integer. The number a 0 is the constant coefficient, or the constant term. Note that a polynomial can be of degree zero: it is just a constant function P = a 0. The number a n, the coefficient of the highest power term, is called the leading coefficient. The term a n n is the leading term. It doesn t matter whether the term is actually written first, last, or anywhere in between. End Behavior What happens when become etremely large or small? Notation:, reads as approaches infinity, means becomes very large in the positive direction. Similarly,, reads as approaches negative infinity, means becomes very large in the negative direction. The end behavior of each particular polynomial function depends only on its leading term both the degree and the leading coefficient: When n is even: If a n > 0, then y as and. If a n < 0, then y as and. When n is odd: If a n > 0, then y as, and y as. If a n < 0, then y as, and y as.
Real Zeros of Polynomials If P is a polynomial and c is a real number, then the following properties are equivalent i.e., either they are all true, or none of them is true. 1. c is a zero of P. 2. = c is a solution of the equation P = 0. 3. c is a factor of P. 4. = c is an -intercept of the graph of P. Behavior Near an -intercept / Shape of the Graph Near a Zero The behavior of a polynomial s graph near each of its zeros, c, depends on the multiplicity of that particular root that is, the number of times c repeats as a root, or, equivalently, the number of time c appear in the factorization of the polynomial. If c is a root of multiplicity m, then the graph takes the general shape of the graph of y = c m near c. To wit m = 1, the usual. m is even, the curve behaves similar to a parabola/ m > 1, m is odd, the curve behaves similar to the graph of y = 3 near the origin.
Intermediate Value Theorem for Polynomials If P is a polynomial function and Pa and Pb have opposite signs, then there eists at least one value c between a and b for which Pc = 0. In other words, between a positive point and a negative point on the graph of a polynomial, there must be at least one root / -intercept. Comment: This really is just a special case of the general Intermediate Value property, which is possessed by every continuous function. An important consequence, for our purpose, of this theorem is that the curve of a polynomial s graph always resides on one side of the -ais between its -intercepts. Polynomial functions are very easy and quick to graph by hand, especially if all the roots of a polynomial are real numbers. Guidelines for Graphing Polynomial Functions Locating Zeros Find all real roots of the polynomial, they are the - intercepts. Test Points End Behavior Graph Break up the real line into intervals using the real roots as endpoints. Test a point from each interval to determine if the graph is above or below the -ais. Determined the end behavior as and. Sketch a smooth continuous curve that obeys the end behavior and that passes through each zero ehibiting the correct behavior according to the zero s multiplicity.
Eample: P = 3 2 2 Eample: P = + 2 1 2 4
Eample: P = 3.01 2 Eample: P = 3 2.01
4.2 Dividing Polynomials Division Algorithm Suppose P and D are polynomials, with D 0, then there eist unique polynomials Q and R such that P = D Q + R. The remainder, R, is either 0 or is a polynomial with degree strictly less than the degree of the divisor, D. Equivalently, it says that the following equality: D R Q D R Q D D P + = + = How to Divide Polynomials: Polynomial Long Division Eample: #11 2 4 3 + 9 2 2 + 4 Eample: #20 Find the quotient and remainder of 8 6 4 13 7 2 2 4 5 +
Synthetic Division When it works, it is faster than long division. But it works only when the divisor is in the form c. Remainder Theorem If the polynomial P is divided by c, then the remainder is the value Pc. Factor Theorem c is a zero of polynomial P if and only if c is a factor of P. Eample: Given that = 1 is a root, completely factor 3 6 2 + 11 6.
4.5 Rational Functions A rational function is a function of the form r = where P and Q are polynomials. P Q A Simple the Simplest? Rational Function [See net page] f = 1 Indeed, using transformations we have learned, any rational function of the form a+ b 1 r = can be obtained from the graph of f =. c+ d Eample: 4 r = + 2
Graph of y 1 =. Domain: All real numbers ecept 0. Range: All real numbers ecept 0. Vertical asymptote: = 0. Horizontal asymptote: y = 0.
If a 0, the epression can be simplified by polynomial division first. Eample: 2 5 r = 1
Asymptotes of Rational Functions A rational function has a vertical asymptote at each zero of its denominator, after simplification to cancel out any common factors shared by its numerator and denominator. For an eample of what happens when there is an un-cancelled common factor, see the graph of y = 2 1/ 1: There are as many vertical asymptotes as the denominator has distinct real roots, if any. Comple roots do not result in vertical asymptotes. The horizontal asymptote occurs when y approaches a finite value as approaches ±. No function could have more than 2 horizontal asymptotes one in each direction, if eist and if different. A rational function, however, can have at most one horizontal asymptote.
In general, the asymptotes of rational functions can be summarized in the following set of rules you will learn about them in the calculus class: P Let r = be a rational function where Q P is a polynomial of degree m and Q is a polynomial of degree n. Further, assume that r has been simplified such that P and Q share no common factors other than perhaps a constant. Vertical Asymptotes: The vertical asymptotes are the lines = a, where a is a zero of the denominator Q. Horizontal Asymptote: i. If m < n, then r has horizontal asymptote y = 0. a ii. If m = n, then r has horizontal asymptote y= b where a and b are, respectively, the leading coefficients of P and Q. iii. If m > n, then r has no horizontal asymptote.
Guidelines for Graphing Rational Functions Factor Factor both the numerator and denominator. Intercepts Find all real roots of the numerator, they are the - intercepts. The y-intercept is r0. Vertical Asymp. The zeros of the denominators are the vertical asymptotes. Plot test points to determine the behavior of the curve whether y or y on either side of each asymptote. Horiz. Asymp. Graph Determined the end behavior as and, to find the eistence and location of the horizontal asymptote. Sketch the graph using the information gathered in the previous steps. Plot additional points as needed to fill in the detail.
Eample: r = 2 2 + 2 3 Eample: 2 r = 2 3 4 + 5 6
Graph of 2 = 2 3+ 4 + 2 3 r : The graph has no -intercepts. What is the mathematical significance of this fact? Note that the curve crosses over the horizontal asymptote just right of = 1. While the graph of a function can never cross a vertical asymptote, it can and does cross a horizontal asymptote. Recall that the horizontal asymptote is determined only by the behavior of the function as and/or. Therefore, the behavior of the function elsewhere has no bearing on the eistence of a horizontal asymptote.
Slant Asymptotes Asymptotes need not to be vertical or horizontal. For eample the ones possessed by the hyperbola 2 y 2 = 1. An asymptote that is neither vertical nor horizontal is called a slant asymptote or oblique asymptote. For rational functions, it eists on the graph whenever the degree of the numerator is eactly one higher than the degree of the denominator. Eample: f = 2 + 2+ 1 Dividing 2 1 + 2 + 1 by, we get f = + 2 +. Therefore, the graph has an oblique asymptote y = + 2. It also has a vertical asymptote = 0. Its graph: