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Factoring a Polynomial

Factoring a Polynomial The Linear Factorization Theorem shows that you can write any n th-degree polynomial as the product of n linear factors. f (x) = an(x – c1)(x – c2)(x – c3) . . . (x – cn) However, this result includes the possibility that some of the values of ci are complex.

Factoring a Polynomial The following theorem says that even if you do not want to get involved with “complex factors,” you can still write f (x) as the product of linear and/or quadratic factors.

Factoring a Polynomial A quadratic factor with no real zeros is said to be prime or irreducible over the reals. Note that this is not the same as being irreducible over the rationals. For example, the quadratic x2 + 1 = (x – i )(x + i ) is irreducible over the reals (and therefore over the rationals). On the other hand, the quadratic x2 – 2 = is irreducible over the rationals but reducible over the reals.

Application

Example 11 – Using a Polynomial Model You are designing candle making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle. You want the height of the candle to be 2 inches less than the length of each side of the candle’s square base. What should the dimensions of your candle mold be? Solution: The volume of a pyramid is where B is the area of the base and h is the height. The area of the base is x2 and the height is (x – 2).

Example 11 – Solution cont’d So, the volume of the pyramid is Substituting 25 for the volume yields the following.

Example 11 – Solution cont’d The possible rational solutions are Use synthetic division to test some of the possible solutions. Note that in this case it makes sense to test only positive x-values. Using synthetic division, you can determine that x = 5 is a solution.

Example 11 – Solution cont’d The other two solutions that satisfy x2 + 3x + 15 = 0 are imaginary and can be discarded. You can conclude that the base of the candle mold should be 5 inches by 5 inches and the height should be 5 – 2 = 3 inches.

Application Before concluding this section, here is an additional hint that can help you find the real zeros of a polynomial. When the terms of f(x) have a common monomial factor, it should be factored out before applying the tests in this section. For instance, by writing you can see that x = 0 is a zero of f and that the remaining zeros can be obtained by analyzing the cubic factor.

Copyright © Cengage Learning. All rights reserved. 2.6 Rational Functions Copyright © Cengage Learning. All rights reserved.

Objectives Find the domains of rational functions. Find the vertical and horizontal asymptotes of the graphs of rational functions. Sketch the graphs of rational functions. Sketch the graphs of rational functions that have slant asymptotes. Use rational functions to model and solve real-life problems.

Introduction

Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial.

Introduction In general, the domain of a rational function of x includes all real numbers except x-values that make the denominator zero. Much of the discussion of rational functions will focus on their graphical behavior near the x-values excluded from the domain.

Example 1 – Finding the Domain of a Rational Function Find the domain of and discuss the behavior of f near any excluded x-values. Solution: Because the denominator is zero when x = 0 the domain of f is all real numbers except x = 0.

Example 1 – Solution cont’d To determine the behavior of f near this excluded value, evaluate f (x) to the left and right of x = 0, as indicated in the following tables.

Example 1 – Solution cont’d Note that as x approaches 0 from the left, f (x) decreases without bound. In contrast, as x approaches 0 from the right, f (x) increases without bound. The graph of f is shown below.

Vertical and Horizontal Asymptotes

Vertical and Horizontal Asymptotes In Example 1, the behavior of f near x = 0 is denoted as follows. The line x = 0 is a vertical asymptote of the graph of f, as shown in Figure 2.21. Figure 2.21

Vertical and Horizontal Asymptotes From this figure, you can see that the graph of f also has a horizontal asymptote—the line y = 0. The behavior of f near y = 0 is denoted as follows.

Vertical and Horizontal Asymptotes Eventually (as x  or x  ), the distance between the horizontal asymptote and the points on the graph must approach zero.

Vertical and Horizontal Asymptotes Figure 2.22 shows the vertical and horizontal asymptotes of the graphs of three rational functions. (a) (b) (c) Figure 2.22

Independent Practice Section 2.5 (page 164) # 113, 131 Section 2.6 (page 177) # 1 – 4 (vocabulary), 7, 8, 71