/*! 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 19 Simplify the rational expression... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 19

Simplify the rational expression. $$ \frac{3(x+2)(x-1)}{6(x-1)^{2}} $$

Short Answer

Expert verified
\( \frac{x+2}{2(x-1)} \)

Step by step solution

01

Factor the Numerator and the Denominator

In the expression \( \frac{3(x+2)(x-1)}{6(x-1)^2} \), the numerator is already factored as \( 3(x+2)(x-1) \). The denominator can be factored as \( 6(x-1)^2 \).
02

Cancel Common Factors

Observe the common factors in the numerator and the denominator. Both have \((x-1)\), so cancel out one \((x-1)\) from the numerator and one from the denominator, simplifying the expression to \( \frac{3(x+2)}{6(x-1)} \).
03

Simplify the Coefficient Fraction

Now, simplify the coefficients \( \frac{3}{6} \) to \( \frac{1}{2} \). Thus, the expression becomes \( \frac{1(x+2)}{2(x-1)} \), which simplifies further to \( \frac{x+2}{2(x-1)} \).

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 Expressions
Factoring is a crucial step in simplifying rational expressions. It's the process where we break down expressions into simpler terms that, when multiplied, give back the original expression. In our exercise, the expression is already factored:
  • The numerator: \(3(x+2)(x-1)\) consists of a constant \(3\) and the factors \((x+2)\) and \((x-1)\).
  • The denominator: \(6(x-1)^2\) includes the constant \(6\) and a repeated factor \((x-1)(x-1)\).
By recognizing these factors, we set the stage for simplifying the expression further. Always ensure each expression is fully factored to identify all possible common factors.
Canceling Common Factors
Once you've factored both the numerator and the denominator, look for common factors. These are terms that appear in both the numerator and the denominator. In our example:
  • The common factor: \((x-1)\) appears once in the numerator and twice in the denominator.
By canceling out one \((x-1)\) from both the numerator and the denominator, we reduce the expression to a simpler form. It's like reducing the size of a fraction by dividing both its top and bottom by the same number. Be careful not to cancel out non-common terms. This step is vital to ensure you do not alter the value of the expression.
Simplifying Fractions
Simplifying fractions involves reducing the expression to its most simplified form. After canceling common factors, the expression from our example becomes \( \frac{3(x+2)}{6(x-1)} \). Here, we focus on reducing the coefficient terms:
  • The coefficient fraction \(\frac{3}{6}\) simplifies to \(\frac{1}{2}\).
Simplifying numerical fractions like this helps in achieving a cleaner expression. Thus, the expression further simplifies to \( \frac{x+2}{2(x-1)} \). When simplifying, always check if there are any other terms - numerical or variable-based - that can be reduced without altering the expression's value.

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

State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.) $$ \frac{b}{b-c}=1-\frac{b}{c} $$

Rationalize the numerator. $$ \frac{\sqrt{3}+\sqrt{5}}{2} $$

Rationalize the denominator. $$ \frac{2}{3-\sqrt{5}} $$

Perform the addition or subtraction and simplify. $$ \frac{x}{x-4}-\frac{3}{x+6} $$

Simplify the compound fractional expression. $$ \frac{\frac{x+2}{x-1}-\frac{x-3}{x-2}}{x+2} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.