Textbook Question
Evaluate each exponential expression in Exercises 1–22. 4−3
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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 11
Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
Verified step by step guidance1
Start by factoring both the numerator and the denominator of the rational expression. For the numerator \(y^2 + 7y - 18\), find two numbers that multiply to \(-18\) and add to \(7\). For the denominator \(y^2 - 3y + 2\), find two numbers that multiply to \(2\) and add to \(-3\).
Rewrite the numerator and denominator as products of their factors. For example, the numerator will be factored into the form \((y + a)(y + b)\) and the denominator into \((y + c)(y + d)\), where \(a\), \(b\), \(c\), and \(d\) are the numbers found in the previous step.
Simplify the rational expression by canceling out any common factors that appear in both the numerator and the denominator.
Identify the values of \(y\) that make the original denominator equal to zero, since these values are excluded from the domain. Set the original denominator equal to zero and solve for \(y\).
State the simplified rational expression along with the domain restrictions, which exclude the values of \(y\) found in the previous step.
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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 the numerator and denominator and then canceling out common factors. This process reduces the expression to its simplest form, making it easier to analyze or use in further calculations.
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Factoring Quadratic Expressions
Factoring quadratics means rewriting a quadratic polynomial as a product of two binomials. Recognizing patterns like factoring trinomials into (y + m)(y + n) where m and n multiply to the constant term and add to the linear coefficient is essential for simplifying rational expressions.
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Domain Restrictions in Rational Expressions
The domain of a rational expression excludes values that make the denominator zero, as division by zero is undefined. Identifying these excluded values requires setting the denominator equal to zero and solving for the variable to find restrictions on the domain.
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Related Practice
Textbook Question
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). x2-3(x-y), for x=8 and y=2
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Textbook Question
In Exercises 1–10, factor out the greatest common factor. x2(2x+5)+17(2x+5)
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Textbook Question
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √144+√25
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Textbook Question
In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (17x3−5x2+4x−3)−(5x3−9x2−8x+11)
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Textbook Question
Express the distance between the numbers -17 and 4 using absolute value. Then evaluate the absolute value.
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