/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Reduce each rational expression ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 7

Reduce each rational expression to lowest terms. $$ \frac{3 x+9}{x^{2}-9} $$

Short Answer

Expert verified
The reduced form of the rational expression is \( \frac{3}{x - 3} \).

Step by step solution

01

Factorize the Numerator

The numerator is given as \( 3x + 9 \). Factor out the common factor in the numerator.\[ 3x + 9 = 3(x + 3) \]
02

Factorize the Denominator

The denominator is given as \( x^2 - 9 \). Recognize this expression as a difference of squares and factorize it accordingly.\[ x^2 - 9 = (x + 3)(x - 3) \]
03

Write the Expression with Factored Form

Now rewrite the rational expression using the factored forms of the numerator and the denominator.\[ \frac{3(x + 3)}{(x + 3)(x - 3)} \]
04

Cancel Common Factors

Identify and cancel out the common factors in the numerator and denominator. Here, \(x + 3\) is a common factor.\[ \frac{3 \textbf{(x + 3)}}{(x + 3)(x - 3)} = \frac{3}{x - 3} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Factorization
Factorization is a fundamental skill in algebra. It's all about breaking down complex expressions into simpler components called factors. For example, in the numerator of our exercise, we have the expression \(3x + 9\). By factoring out the common factor, we get:
  • Identify the greatest common factor (GCF) of the terms.
  • In \(3x + 9\), the GCF is 3.
  • We factor out the 3 to get \(3(x + 3)\).
Simplifying the numerator makes it easier to work with the entire rational expression. Practice different factorization techniques to get comfortable with breaking down expressions.
Difference of Squares
The difference of squares is a special factoring technique. It applies when you have an expression that looks like \(a^2 - b^2\). This can be factored into \((a + b)(a - b)\). Let's look at our exercise. The denominator given is \(x^2 - 9\). Recognize this as a difference of squares:
  • Rewrite 9 as \(3^2\).
  • Now you have \(x^2 - 3^2\).
  • Apply the difference of squares formula so \(x^2 - 9 = (x + 3)(x - 3)\).
This technique is useful for simplifying rational expressions and finding common factors.
Canceling Common Factors
Simplifying rational expressions often involves canceling common factors. After factorization, you might find terms that appear in both the numerator and the denominator. For our example, the factored expression is \(\frac{3(x + 3)}{(x + 3)(x - 3)}\). Notice that \(x + 3\) is present in both the numerator and denominator.
  • Common factors can be divided out or 'canceled'.
  • This simplifies the expression to \(\frac{3}{x - 3}\).
Canceling common factors makes expressions simpler and more manageable. Just ensure you never cancel terms that may equal zero, as this would make the original expression undefined.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$\left(x^{2}+4\right)^{4 / 3}+x \cdot \frac{4}{3}\left(x^{2}+4\right)^{1 / 3} \cdot 2 x$$

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$8 x^{1 / 3}-4 x^{-2 / 3} \quad x \neq 0$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-\sqrt{3}}{\sqrt{8}}$$

Simplify each expression. Assume that all variables are positive when they appear. $$(3 \sqrt{6})(2 \sqrt{2})$$

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt[3]{-40 x^{14} y^{10}}$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.