Textbook Question
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−8x2+x+2; between 2 and 3
Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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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 41
Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x3/(3x2+1)
Verified step by step guidance1
Identify the degrees of the numerator and denominator polynomials in the rational function \(h(x) = \frac{12x^{3}}{3x^{2} + 1}\). The degree of the numerator is 3, and the degree of the denominator is 2.
Recall the rule for horizontal asymptotes of rational functions:
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
- If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique/slant asymptote instead).
Since the degree of the numerator (3) is greater than the degree of the denominator (2), conclude that there is no horizontal asymptote for \(h(x)\).
Optionally, to further analyze the behavior of \(h(x)\) as \(x\) approaches infinity, consider performing polynomial long division or examining the end behavior to find any oblique asymptotes.
Summarize: For \(h(x) = \frac{12x^{3}}{3x^{2} + 1}\), no horizontal asymptote exists because the numerator's degree is higher than the denominator's.
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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 the degrees and leading coefficients of the numerator and denominator polynomials is essential for analyzing the behavior of the function, especially for asymptotes.
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Horizontal Asymptotes
Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator polynomials and indicates the value the function approaches at extreme x-values.
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Degree Comparison of Polynomials
The degree of a polynomial is the highest power of x in the expression. Comparing the degrees of the numerator and denominator in a rational function helps determine the horizontal asymptote: if numerator degree < denominator degree, asymptote is y=0; if equal, ratio of leading coefficients; if numerator degree > denominator degree, no horizontal asymptote.
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05:16Standard Form of Polynomials
Related Practice
Textbook Question
For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros.
Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation.
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Textbook Question
Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x4+10x3+5x2+x+1; f(−2/3)
Textbook Question
An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=−4x2+8x−3
Textbook Question
Use synthetic division to divide f(x)=x3−4x2+x+6 by x+1. Use the result to find all zeros of f.