Textbook Question
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
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Ch. P - Fundamental Concepts of Algebra
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 1, Problem 16
Multiply or divide as indicated.
Verified step by step guidance1
Start by writing the expression clearly: \(\frac{6x+9}{3x-15} \cdot \frac{x-5}{4x+6}\).
Factor all polynomials in the numerators and denominators where possible. For example, factor out the greatest common factor (GCF) from each: \(6x+9 = 3(2x+3)\), \(3x-15 = 3(x-5)\), and \(4x+6 = 2(2x+3)\).
Rewrite the expression using the factored forms: \(\frac{3(2x+3)}{3(x-5)} \cdot \frac{x-5}{2(2x+3)}\).
Look for common factors in the numerators and denominators across the entire expression that can be canceled out. Cancel these common factors to simplify the expression.
After canceling, multiply the remaining numerators together and the remaining denominators together to write the simplified product as a single fraction.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Polynomials
Factoring involves rewriting polynomials as products of simpler expressions. This is essential for simplifying rational expressions by canceling common factors. For example, 6x + 9 can be factored as 3(2x + 3), and 3x - 15 as 3(x - 5).
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Introduction to Factoring Polynomials
Multiplication of Rational Expressions
Multiplying rational expressions involves multiplying the numerators together and the denominators together. After multiplication, the expression should be simplified by factoring and canceling common factors to reduce it to simplest form.
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02:58Rationalizing Denominators
Simplifying Rational Expressions
Simplifying rational expressions means reducing them to their simplest form by canceling common factors in the numerator and denominator. This requires identifying and factoring expressions fully to see which terms can be canceled.
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Related Practice
Textbook Question
In Exercises 11–16, factor by grouping. x3+6x2−2x−12
Textbook Question
The formula C=5/9(F-32) expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17–18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale. 50 °F
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Textbook Question
In Exercises 15–58, find each product. (x+1)(x2−x+1)
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Textbook Question
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). (2x+3y)/(x+1), for x=-2 and y=4
Textbook Question
Evaluate each exponential expression in Exercises 1–22.