Textbook Question
Find k so that 4x+3 is a factor of 20x^3+23x^2-10x+k.
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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 75
Follow the seven steps to graph each rational function. f(x)=x4/(x2+2)
Verified step by step guidance1
Identify the domain of the function \(f(x) = \frac{x^{4}}{x^{2} + 2}\). Since the denominator is \(x^{2} + 2\), which is always positive for all real \(x\), the domain is all real numbers, \((-\infty, \infty)\).
Find the intercepts: For the y-intercept, evaluate \(f(0) = \frac{0^{4}}{0^{2} + 2}\). For the x-intercepts, set the numerator equal to zero, \(x^{4} = 0\), and solve for \(x\).
Determine the vertical asymptotes by finding values of \(x\) that make the denominator zero. Since \(x^{2} + 2 = 0\) has no real solutions, there are no vertical asymptotes.
Find the horizontal or oblique asymptotes by comparing the degrees of the numerator and denominator. The numerator degree is 4 and the denominator degree is 2. Since the numerator degree is greater, perform polynomial division of \(x^{4}\) by \(x^{2} + 2\) to find the oblique asymptote.
Analyze the end behavior of the function using the quotient from the polynomial division and sketch the graph accordingly, including the intercepts and asymptotes found.
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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), where Q(x) ≠ 0. Understanding the domain, zeros, and behavior of both numerator and denominator is essential for graphing and analyzing these functions.
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Intro to Rational Functions
Domain and Vertical Asymptotes
The domain of a rational function excludes values that make the denominator zero. These values often correspond to vertical asymptotes, where the function approaches infinity or negative infinity, indicating important features in the graph.
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3:12Determining Vertical Asymptotes
End Behavior and Horizontal/Oblique Asymptotes
End behavior describes how the function behaves as x approaches infinity or negative infinity. For rational functions, this is determined by comparing the degrees of numerator and denominator, which helps identify horizontal or oblique asymptotes guiding the graph's long-term trend.
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4:48Determining Horizontal Asymptotes
Related Practice
Textbook Question
In Exercises 69–74, solve each inequality and graph the solution set on a real number line. (x + 3)/(x - 4) ≤ 5
Textbook Question
In Exercises 73–74, use the graph of the rational function to solve each inequality.
1/4(x + 2) > - 3/4(x - 2)
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.
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Textbook Question
In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=(x2+x−12)/(x2−4)
Textbook Question
When 2x2−7x+9 is divided by a polynomial, the quotient is 2x-3 and the remainder is 3. Find the polynomial.
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