/*! 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 15 Simplify each rational expressio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 15

Simplify each rational expression. See Examples 2 through \(5 .\) $$ \frac{x^{2}-9}{3+x} $$

Short Answer

Expert verified
\( x - 3 \)

Step by step solution

01

Identify the Numerator and Denominator

The given rational expression is \( \frac{x^{2} - 9}{3 + x} \). Here, the numerator is \( x^2 - 9 \) and the denominator is \( 3 + x \).
02

Factor the Numerator

The numerator \( x^2 - 9 \) is a difference of squares and can be factored as \( (x - 3)(x + 3) \).
03

Substitute the Factored Numerator

Rewrite the expression using the factored numerator: \( \frac{(x - 3)(x + 3)}{3 + x} \).
04

Simplify the Expression

Recognize that \( 3 + x \) is the same as \( x + 3 \). Therefore, the expression simplifies to \( x - 3 \) since \( x + 3 \) cancels out in the numerator and denominator.

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.

Factoring Polynomials
Polynomials can often be expressed in a simpler form through factoring. Factoring involves writing a polynomial as the product of its factors. By breaking down a polynomial to its basic components, we can often simplify and solve it more easily.
To factor a polynomial, look for common factors, patterns, or special identities. Observe if the expression fits known identities like the difference of squares, perfect square trinomials, or the sum/difference of cubes.
  • A polynomial like \(x^2 - 9\) is recognized as a difference of squares.
  • This means it can be expressed as \((x - 3)(x + 3)\).
Understanding and practicing factoring can greatly ease the manipulation of polynomials and simplifying complex expressions.
Difference of Squares
The difference of squares is a specific form of polynomial expression and a key concept when simplifying rational expressions. It applies to situations where you have a subtraction between two squared terms.
A polynomial is a difference of squares if it can be expressed in the form \(a^2 - b^2\). This can be factored into \((a - b)(a + b)\).
  • In the expression \(x^2 - 9\), identify both terms as perfect squares.
  • The term \(x^2\) is a square of \(x\) and \(9\) is a square of \(3\).
By recognizing and applying this rule, you can easily factor expressions like \(x^2 - 9\) into \((x - 3)(x + 3)\), simplifying the process of working with polynomials.
Numerator and Denominator Identification
One of the first steps in simplifying a rational expression is identifying its numerator and denominator. A rational expression has the general form \(\frac{P}{Q}\), where \(P\) is the numerator and \(Q\) is the denominator.
In the given example, \(\frac{x^2 - 9}{3 + x}\), determining these parts correctly is crucial because:
  • The simplification process often involves operations on both the numerator and the denominator.
  • Here, the numerator is \(x^2 - 9\), which can be factored using the difference of squares method.
  • The denominator is \(3 + x\), which may sometimes be simplified by matching it to a form present in the numerator.
Understanding and identifying these components allow you to approach the expression logically and apply necessary factoring or cancellation methods for simplification.

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

Factor the following. $$ a^{3}-27 $$

If the area of a parallelogram is \(\left(x^{4}-23 x^{2}+9 x-5\right)\) square centimeters and its base is \((x+5)\) centimeters, find its height.

For the given polynomial \(P(x)\) and the given \(c,\) use the remainder theorem to find \(P(c)\). $$ P(x)=x^{5}+x^{4}-x^{3}+3 ; \frac{1}{2} $$

Divide. $$ \left(5 x^{4}-5 x^{2}+10 x^{3}-10 x\right) \div(5 x+10) $$

For each statement, find the constant of variation and the variation equation. See Examples 5 and 6. \(y\) varies jointly as \(x\) and the cube of \(z ; y=120\) when \(x=5\) and \(z=2\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.