Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. 2x2 + 5x - 3 ≤ 0
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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 89
In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. 5x2/(x2−4) ⋅ (x2+4x+4)/(10x3)
Verified step by step guidance1
Identify the given expression: \( \frac{5x^2}{x^2 - 4} \cdot \frac{x^2 + 4x + 4}{10x^3} \). Our goal is to multiply these two rational expressions and simplify the result.
Factor all polynomials where possible to simplify the expression. Note that \( x^2 - 4 \) is a difference of squares and factors as \( (x - 2)(x + 2) \). Also, \( x^2 + 4x + 4 \) is a perfect square trinomial and factors as \( (x + 2)^2 \).
Rewrite the expression with factored forms: \( \frac{5x^2}{(x - 2)(x + 2)} \cdot \frac{(x + 2)^2}{10x^3} \).
Multiply the numerators together and the denominators together: numerator \( = 5x^2 \cdot (x + 2)^2 \), denominator \( = (x - 2)(x + 2) \cdot 10x^3 \).
Simplify the expression by canceling common factors. For example, \( (x + 2) \) appears in both numerator and denominator, and powers of \( x \) can be reduced. After simplification, write the simplified expression for \( f(x) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simplifying Rational Expressions
Simplifying rational expressions involves factoring polynomials in the numerator and denominator and then canceling common factors. This process reduces the expression to its simplest form, making it easier to analyze and graph.
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Multiplication of Rational Expressions
When multiplying rational expressions, multiply the numerators together and the denominators together. After multiplication, simplify the resulting expression by factoring and canceling common terms to obtain the simplest form.
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Graphing Rational Functions
Graphing rational functions requires understanding their domain, intercepts, asymptotes, and behavior near undefined points. Simplifying the function first helps identify vertical and horizontal asymptotes and key points for an accurate graph.
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Related Practice
Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. (x - 4)/(x - 1) ≤ 0
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)=(x3+1)/(x2+2x)
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Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)
Textbook Question
Solve each inequality in Exercises 86–91 using a graphing utility. x3 + x2 - 4x - 4 > 0
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Textbook Question
In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. x/(2x+6) − 9/(x2−9)