교과서 질문
Solve each inequality in Exercises 86–91 using a graphing utility. (x - 4)/(x - 1) ≤ 0
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Ch. 3 - Polynomial and Rational Functions
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
4장, 문제 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)
검증된 단계별 안내1
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 \( f(x) \).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
19m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
05:07
Simplifying Algebraic Expressions
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.
추천 영상:
02:58Rationalizing Denominators
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.
추천 영상:
8:19How to Graph Rational Functions
관련 실천
교과서 질문
In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x3+1)/(x2+2x)
601
views
교과서 질문
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. (1 − 3/(x+2)) / (1 + 1/(x−2))
793
views
교과서 질문
Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)
527
views
교과서 질문
Solve each inequality in Exercises 86–91 using a graphing utility. x3 + x2 - 4x - 4 > 0
548
views
교과서 질문
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. x/(2x+6) − 9/(x2−9)
833
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