/*! 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 25 Multiply. $$\left(x+\frac{2}{3... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 25

Multiply. $$\left(x+\frac{2}{3}\right)\left(x-\frac{1}{3}\right)$$

Short Answer

Expert verified
\(x^2 + \frac{1}{3}x - \frac{2}{9}\).

Step by step solution

01

Expand the Expression Using the Distributive Property

Multiply each term in the first binomial \( (x + \frac{2}{3}) \) by each term in the second binomial \( (x - \frac{1}{3}) \). Use the formula \((a+b)(c+d)=ac+ad+bc+bd\).
02

Calculate Each Multiplication

Compute the products individually: \( x \cdot x = x^2 \), \( x \cdot -\frac{1}{3} = -\frac{1}{3}x \), \( \frac{2}{3} \cdot x = \frac{2}{3}x \), and \( \frac{2}{3} \cdot -\frac{1}{3} = -\frac{2}{9} \).
03

Combine Like Terms

Add the terms obtained from multiplication to simplify: \( x^2 - \frac{1}{3}x + \frac{2}{3}x - \frac{2}{9} \). Combine \(-\frac{1}{3}x\) and \(\frac{2}{3}x\) to get \(\frac{1}{3}x\).
04

Write the Final Simplified Expression

Substitute the simplified terms back: \( x^2 + \frac{1}{3}x - \frac{2}{9} \).

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.

Binomials
In algebra, binomials are expressions containing two terms that are typically separated by a plus or minus sign. Each term can be a number, a variable, or their combination. For example, in the expression \(x + \frac{2}{3}\), the terms are \(x\) and \(\frac{2}{3}\). Similarly, \(x - \frac{1}{3}\) is also a binomial with \(x\) and \(-\frac{1}{3}\) as its terms.

Understanding binomials is an essential foundation in algebra as they often appear in equations and expressions. When multiplying binomials, you apply the distributive property to simplify or expand them.
Distributive Property
The distributive property is a valuable algebraic tool that helps expand expressions for better manipulation and simplicity. In simple terms, it allows one to distribute multiplication over addition or subtraction inside an expression. This property can be summed up with the formula: \((a+b)(c+d)=ac+ad+bc+bd\).

To apply it to our expression \( (x + \frac{2}{3})(x - \frac{1}{3}) \):

  • Multiply \(x\) by \(x\)
  • Multiply \(x\) by \(-\frac{1}{3}\)
  • Multiply \(\frac{2}{3}\) by \(x\)
  • Multiply \(\frac{2}{3}\) by \(-\frac{1}{3}\)
This process efficiently rearranges and simplifies the original expression, preparing it for further calculation or interpretation.
Simplifying Expressions
Simplifying expressions involves combining like terms to create a more concise and understandable form of the expression. In our example, we are simplifying \(x^2 - \frac{1}{3}x + \frac{2}{3}x - \frac{2}{9}\).

Here are the steps to simplify:

  • Identify like terms, which are terms with the same variable factor. In this case, \(-\frac{1}{3}x\) and \(\frac{2}{3}x\) are like terms.
  • Combine these like terms: \(-\frac{1}{3}x + \frac{2}{3}x = \frac{1}{3}x\).
  • Write the expression with combined terms: \(x^2 + \frac{1}{3}x - \frac{2}{9}\).
Simplifying makes an equation clearer and easier to interpret or solve further.

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

Multiply each expression. $$ 2 x\left(x^{2}+7 x-5\right) $$

Multiply. See Section 5.1 $$ 6 r^{3}\left(7 r^{10}\right) $$

The volume of the swimming pool shown is \(\left(36 x^{5}-12 x^{3}\right.\) \(\left.+6 x^{2}\right)\) cubic feet. If its height is \(2 x\) feet and its width is \(3 x\) feet, find its length.

Add or subtract as indicated. See Examples 9 through \(13 .\) $$ \left(7 a^{2}-3 b^{2}+10\right)-\left(-2 a^{2}+b^{2}-12\right) $$

Explain how to check a polynomial long division result when the remainder is 0
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and 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.