/*! 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 38 Write each rational expression i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 38

Write each rational expression in lowest terms. $$ \frac{m^{2}-25}{4 m-20} $$

Short Answer

Expert verified
\( \frac{m + 5}{4} \).

Step by step solution

01

Identify the Numerator and Denominator

The given rational expression is \( \frac{m^{2}-25}{4m-20} \). Here, the numerator is \( m^2 - 25 \), and the denominator is \( 4m - 20 \).
02

Factor the Numerator

The numerator \( m^2 - 25 \) is a difference of squares. It can be factored as \( (m + 5)(m - 5) \).
03

Factor the Denominator

The denominator \( 4m - 20 \) can be factored by taking out the greatest common factor (GCF). The GCF is 4, so \( 4m - 20 \) can be written as \( 4(m - 5) \).
04

Simplify the Expression

The rational expression \( \frac{(m + 5)(m - 5)}{4(m - 5)} \) has a common factor of \( m - 5 \) in both the numerator and the denominator. Cancel out \( m - 5 \) to get \( \frac{m + 5}{4} \).

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 simplified by breaking them down into simpler factors.
Factoring involves expressing a polynomial as a product of its factors.
For instance, in the numerator \( m^2 - 25 \), we notice it's a difference of squares, which simplifies to \( (m + 5)(m - 5) \).
Being able to identify special patterns like difference of squares or factoring by grouping makes the process easier.
Greatest Common Factor
The greatest common factor (GCF) is the largest factor shared by two or more terms.
To simplify rational expressions, you should factor out the GCF from both the numerator and the denominator.
In our example, the denominator is \( 4m - 20 \). The GCF of 4 and 20 is 4.
Factoring out the GCF, we get \( 4(m - 5) \). Recognizing and factoring out the GCF can simplify complex polynomials into simpler terms.
Difference of Squares
A difference of squares is a specific type of polynomial.
It can be identified by the form \(a^2 - b^2\). This can always be factored into \((a + b)(a - b)\).
In our example, \( m^2 - 25 \) is a difference of squares because it can be written as \( m^2 - 5^2 \).
Thus, it factors to \( (m + 5)(m - 5) \). Recognizing the difference of squares makes the factorization process quicker and simpler.
Canceling Out Common Factors
Once both the numerator and denominator are factored, you might find common factors.
In our simplified expression \( \frac{(m + 5)(m - 5)}{4(m - 5)} \), both the numerator and denominator have the factor \(m - 5 \).
Canceling out these common factors leaves us with a more straightforward expression. In this case, \( \frac{m + 5}{4} \).
Always cancel out common factors to reduce rational expressions to their lowest terms.

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

Solve each problem mentally. Use proportions in Exercises 23 and 24. In a mathematics class, 3 of every 4 students are girls. If there are 28 students in the class, how many are girls? How many are boys?

Work each problem. A concours d'elegance is a competition in which a maximum of 100 points is awarded to a car on the basis of its general attractiveness. The function defined by the rational expression $$ c(x)=\frac{1010}{49(101-x)}-\frac{10}{49} $$ approximates the cost, in thousands of dollars, of restoring a car so that it will win \(x\) points. (a) Simplify the expression for \(c(x)\) by performing the indicated subtraction. (b) Use the simplified expression to determine how much it would cost to win 95 points.

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{11 x-13}{2 x-3}-\frac{3 x-1}{2 x-3} $$

Multiply or divide as indicated. $$ \frac{12 x-10 y}{3 x+2 y} \cdot \frac{6 x+4 y}{10 y-12 x} $$

Multiply or divide as indicated. $$ \frac{a^{2}-1}{4 a} \cdot \frac{2}{1-a} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.