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Chapter 7: Problem 58

Perform the indicated operations. $$ \frac{x^{2}+5 x}{x^{2}-25} \cdot \frac{3 x-15}{x^{2}} $$

Short Answer

Expert verified
The simplified result is \(\frac{3}{x^2}\).

Step by step solution

01

Factorize the Expressions

The first step is to factorize both the numerator and the denominator of each fraction. For the first fraction \(\frac{x^{2}+5x}{x^{2}-25}\), the numerator can be written as \(x(x+5)\) and the denominator can be factored as the difference of squares: \((x+5)(x-5)\). For the second fraction \(\frac{3x-15}{x^2}\), the numerator can be factored out as \(3(x-5)\) and the denominator remains as \(x^2\).
02

Combine and Simplify the Fractions

Now substitute the factorized forms into the combined expression: \(\frac{x(x+5)}{(x+5)(x-5)} \cdot \frac{3(x-5)}{x^2}\). Before multiplying, cancel any common factors between numerators and denominators. \((x+5)\) and \((x-5)\) cancel out from both fractions. The expression simplifies to \(\frac{3x}{x^3}\).
03

Simplify by Cancelling Out Like Terms

The expression \(\frac{3x}{x^3}\) can be simplified by cancelling one \(x\) from both the numerator and the denominator, resulting in \(\frac{3}{x^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Understanding how to factor polynomials is crucial. It involves breaking down a polynomial into simpler components (factors) that multiply to give the original polynomial. Polynomials often need factoring during algebraic fraction operations.
Here are some common factoring techniques:
  • **Greatest Common Factor (GCF):** This method involves identifying the largest polynomial that divides all terms. For example, in the expression \(x^2 + 5x\), the GCF is \(x\), leading to \(x(x+5)\).
  • **Difference of Squares:** This technique applies when a polynomial is a difference between two perfect squares, such as \(x^2 - 25\), which factors to \((x+5)(x-5)\).
Recognizing these patterns and applying them efficiently helps in simplifying algebraic expressions with ease.
Rational Expressions
Rational expressions are fractions where both the numerator and the denominator are polynomials. Working with rational expressions involves similar rules to numerical fractions.
Some important points to keep in mind:
  • **Identifying Restrictions:** Since division by zero is undefined, determine the values making any denominator zero, such as \(x = 5\) for \(x^2 - 25\).
  • **Operations:** Much like numerical fractions, you can add, subtract, multiply, and divide rational expressions but must pay attention to factoring and simplifying.
Simplifying rational expressions often initiates with factoring both the numerators and denominators, as demonstrated in our example.
Simplifying Expressions
Simplifying expressions aims to present them in their simplest form. It involves reducing complexity by canceling common factors or terms.
Here鈥檚 how to tackle simplification:
  • **Cancelling Common Factors:** For the expression \(\frac{x(x+5)}{(x+5)(x-5)} \cdot \frac{3(x-5)}{x^2}\), you identify and cancel \((x+5)\) and \((x-5)\) in the fractions before multiplying.
  • **Power of Variables:** Restructuring terms based on their powers, such as converting \(\frac{3x}{x^3}\) into \(\frac{3}{x^2}\) by cancelling \(x\).
Mastering simplification helps manage complex algebraic problems more effectively, allowing clearer and more manageable solutions.

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Most popular questions from this chapter

Does \(\frac{x}{x+5}\) simplify to \(\frac{1}{5} ?\) Why or why not?

Solve the following linear and quadratic equations. See Sections 2.3 and 6.6 $$ 2(3 x+1)+15=-7 $$

Perform each indicated operation. See Section 1.3. $$ \frac{7}{8} \div \frac{1}{2} $$

Perform the indicated operations. Addition, subtraction, multiplication, and division of rational expressions are included here. $$ \frac{5 a+10}{18} \div \frac{a^{2}-4}{10 a} $$

Perform each indicated operation. $$ \frac{5}{x^{2}-4}+\frac{2}{x^{2}-4 x+4}-\frac{3}{x^{2}-x-6} $$
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