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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 89
Capitolo 4, Problema 89

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. 5x2/(x2−4) ⋅ (x2+4x+4)/(10x3)

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Identify the given expression: \( \frac{5x^2}{x^2 - 4} \cdot \frac{x^2 + 4x + 4}{10x^3} \). Our goal is to multiply these two rational expressions and simplify the result.
Factor all polynomials where possible to simplify the expression. Note that \( x^2 - 4 \) is a difference of squares and factors as \( (x - 2)(x + 2) \). Also, \( x^2 + 4x + 4 \) is a perfect square trinomial and factors as \( (x + 2)^2 \).
Rewrite the expression with factored forms: \( \frac{5x^2}{(x - 2)(x + 2)} \cdot \frac{(x + 2)^2}{10x^3} \).
Multiply the numerators together and the denominators together: numerator \( = 5x^2 \cdot (x + 2)^2 \), denominator \( = (x - 2)(x + 2) \cdot 10x^3 \).
Simplify the expression by canceling common factors. For example, \( (x + 2) \) appears in both numerator and denominator, and powers of \( x \) can be reduced. After simplification, write the simplified expression for \( f(x) \).

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Simplifying Rational Expressions

Simplifying rational expressions involves factoring polynomials in the numerator and denominator and then canceling common factors. This process reduces the expression to its simplest form, making it easier to analyze and graph.
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Multiplication of Rational Expressions

When multiplying rational expressions, multiply the numerators together and the denominators together. After multiplication, simplify the resulting expression by factoring and canceling common terms to obtain the simplest form.
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Graphing Rational Functions

Graphing rational functions requires understanding their domain, intercepts, asymptotes, and behavior near undefined points. Simplifying the function first helps identify vertical and horizontal asymptotes and key points for an accurate graph.
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