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Simplify by reducing the index of the radical.
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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 109
Simplify each rational expression. Also, list all numbers that must be excluded from the domain. [x^3+2x^2]/[x+2]
Guida verificata passo dopo passo1
Factor the numerator \(x^3 + 2x^2\). Start by factoring out the greatest common factor (GCF), which is \(x^2\). This gives \(x^2(x + 2)\).
Identify the denominator \(x + 2\). Note that the denominator cannot be zero, so exclude \(x = -2\) from the domain.
Simplify the rational expression by canceling out the common factor \(x + 2\) in the numerator and denominator. This is valid as long as \(x \neq -2\).
After canceling, the simplified expression is \(x^2\).
Summarize the solution: The simplified expression is \(x^2\), and the excluded value from the domain is \(x = -2\).
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Rational Expressions
A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions involves factoring both parts and reducing them by canceling out common factors. Understanding how to manipulate polynomials is essential for simplifying rational expressions effectively.
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Rationalizing Denominators
Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that when multiplied together give the original polynomial. This is crucial in simplifying rational expressions, as it allows for the identification and cancellation of common factors in the numerator and denominator.
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Domain of a Rational Expression
The domain of a rational expression consists of all the possible values of the variable that do not make the denominator equal to zero. Identifying these values is important because they must be excluded from the domain to avoid undefined expressions. In the given example, finding the values that make the denominator zero is essential for determining the domain.
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02:58Rationalizing Denominators
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