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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 98a
Chapter 1, Problem 98a

Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [10(4x2-9)2 - 25x(4x2-9)3] / [15(4x2-9)6]

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1
Identify the common factors in the numerator and denominator. Notice that both contain powers of the expression \((4x^2 - 9)\).
Rewrite the numerator by factoring out the smallest power of \((4x^2 - 9)\) present in both terms. The numerator is \(10(4x^2 - 9)^2 - 25x(4x^2 - 9)^3\), so factor out \((4x^2 - 9)^2\):
\[ (4x^2 - 9)^2 (10 - 25x(4x^2 - 9)) \]
Simplify inside the parentheses: \(10 - 25x(4x^2 - 9)\).
Rewrite the entire expression as a fraction, then divide numerator and denominator by the common factor \((4x^2 - 9)^2\) to reduce the powers in the denominator:
\[ \frac{(4x^2 - 9)^2 (10 - 25x(4x^2 - 9))}{15(4x^2 - 9)^6} = \frac{10 - 25x(4x^2 - 9)}{15(4x^2 - 9)^{6-2}} = \frac{10 - 25x(4x^2 - 9)}{15(4x^2 - 9)^4} \]

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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 a polynomial as a product of simpler polynomials or expressions. Recognizing common patterns like difference of squares or factoring out the greatest common factor helps simplify complex expressions. In this problem, factoring terms like (4x^2 - 9) is essential to reduce the rational expression.
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Introduction to Factoring Polynomials

Simplifying Rational Expressions

Simplifying rational expressions means reducing fractions by canceling common factors in the numerator and denominator. This requires factoring both parts completely and then dividing out shared factors. It is important to consider the domain restrictions, especially when variables represent positive real numbers.
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Simplifying Algebraic Expressions

Exponent Rules for Expressions

Exponent rules govern how to handle powers of expressions, such as multiplying powers with the same base by adding exponents or factoring powers out of terms. Understanding how to manipulate expressions like (4x^2 - 9)^n is crucial for combining and simplifying terms in the numerator and denominator.
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Introduction to Exponent Rules
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