Textbook Question
Use synthetic division to show that 5 is a solution of x^4−4x^3−9x^2+16x+20=0. Then solve the polynomial equation.
6
views
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 79
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(3x2+x−4)/(2x2−5x)
Verified step by step guidance1
Identify the domain of the function by finding the values of that make the denominator zero. Solve to find these values, since the function is undefined there.
Find the intercepts: For the y-intercept, evaluate if it is in the domain. For the x-intercepts, set the numerator equal to zero and solve to find the values of where .
Determine any vertical asymptotes by using the values of that make the denominator zero (from step 1), provided these values do not also make the numerator zero (which would indicate a hole instead).
Find the horizontal or oblique (slant) asymptote by comparing the degrees of the numerator and denominator polynomials. Since both are degree 2, divide the leading coefficients to find the horizontal asymptote.
Analyze the behavior of the function near the vertical asymptotes and at the ends of the domain by testing values of in each interval determined by the vertical asymptotes, and use this information to sketch the graph.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
17mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Functions
A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined values and to analyze the function's behavior.
Recommended video:
6:04
Intro to Rational Functions
Asymptotes
Asymptotes are lines that the graph of a function approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe end behavior as x approaches infinity or negative infinity.
Recommended video:
6:24Introduction to Asymptotes
Graphing Steps for Rational Functions
Graphing rational functions involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes, then plotting points to sketch the curve. Following a systematic approach ensures an accurate and complete graph.
Recommended video:
8:19How to Graph Rational Functions
Related Practice
Textbook Question
Solve the variation problems in Exercises 77–82. The distance that a body falls from rest is directly proportional to the square of the time of the fall. If skydivers fall 144 feet in 3 seconds, how far will they fall in 10 seconds?
1
views
Textbook Question
Solve the variation problems in Exercises 77–82. The pitch of a musical tone varies inversely as its wavelength. A tone has a pitch of 660 vibrations per second and a wavelength of 1.6 feet. What is the pitch of a tone that has a wavelength of 2.4 feet?
3
views
Textbook Question
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x2−1)/x
Textbook Question
Find the inverse of f(x)=(x−10)/(x+10).
Textbook Question
Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x2+4x−1=0