Domanda del libro di testo
State the name of the property illustrated: (6 • 3) • 9 = 6 • (3 • 9)
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 1, Problema 14
In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (x2−14x+49)/(x2−49)
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Factor both the numerator and the denominator of the rational expression. The numerator, \(x^2 - 14x + 49\), is a perfect square trinomial and can be factored as \((x - 7)^2\). The denominator, \(x^2 - 49\), is a difference of squares and can be factored as \((x - 7)(x + 7)\).
Write the rational expression in its factored form: \(\frac{(x - 7)^2}{(x - 7)(x + 7)}\).
Simplify the rational expression by canceling out one \((x - 7)\) term from the numerator and denominator. The simplified expression becomes \(\frac{x - 7}{x + 7}\).
Identify the values of \(x\) that must be excluded from the domain. These are the values that make the denominator equal to zero. Set \((x - 7)(x + 7) = 0\) and solve for \(x\). The excluded values are \(x = 7\) and \(x = -7\).
State the simplified rational expression and the domain restrictions. The simplified expression is \(\frac{x - 7}{x + 7}\), and the domain excludes \(x = 7\) and \(x = -7\).
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Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions often involves factoring both the numerator and the denominator to identify common factors that can be canceled. Understanding how to manipulate these expressions is crucial for solving problems involving them.
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02:58
Rationalizing Denominators
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of its factors. For example, the expression x^2 - 14x + 49 can be factored into (x - 7)(x - 7) or (x - 7)^2. This process is essential for simplifying rational expressions and identifying restrictions on the variable.
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07:30Introduction to Factoring Polynomials
Domain of a Rational Expression
The domain of a rational expression consists of all real numbers except those that make the denominator equal to zero. In the expression (x^2−14x+49)/(x^2−49), the denominator x^2 - 49 factors to (x - 7)(x + 7), indicating that x cannot be 7 or -7. Identifying these exclusions is vital for understanding the behavior of the expression.
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02:58Rationalizing Denominators
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