/*! 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 102 Simplify. $$ 3 w^{3}+5 w^{3}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 102

Simplify. $$ 3 w^{3}+5 w^{3}-4 w^{3}+12 w^{3}-2 w^{2} $$

Short Answer

Expert verified
The simplified expression is \( 16w^3 - 2w^2 \).

Step by step solution

01

- Group Like Terms

Identify and group the terms with the same variable and exponent. In this expression, that means grouping the terms with the same powers of w: \[ 3w^3, 5w^3, -4w^3, 12w^3 \] and \[-2w^2\].
02

- Combine Like Terms (w^3 terms)

Add or subtract the coefficients of the like terms. For the \(w^3\) terms: \[ 3 + 5 - 4 + 12 = 16 \]So, \( 3w^3 + 5w^3 - 4w^3 + 12w^3 = 16w^3 \).
03

- Rewrite the Expression

Combine your results from the similar terms and write out the simplified expression. The \(w^2\) term remains as it is: \[ 16w^3 - 2w^2 \].

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.

Understanding Like Terms
In polynomial expressions, 'like terms' are terms that have the same variables raised to the same power. For example, in the expression we are simplifying, terms like \( 3w^3 \) and \( 5w^3 \) are like terms because they both have \( w^3 \). On the other hand, \( 3w^3 \) and \( -2w^2 \) are not like terms because one term is \( w^3 \) and the other is \( w^2 \). Identifying like terms is crucial because only like terms can be combined to simplify an expression. Without this step, we can't move forward to simplify the polynomial efficiently.
Combining Coefficients
After identifying the like terms, the next step is to combine their coefficients. Coefficients are the numerical parts of the terms. For instance, in \( 3w^3 \), the coefficient is 3. If we have several like terms, such as \( 3w^3 \), \( 5w^3 \), \( -4w^3 \), and \( 12w^3 \), we combine them by adding or subtracting their coefficients. In this case, we carry out the operation:
  • \( 3 + 5 - 4 + 12 = 16 \)
This tells us that the combined coefficient for the \( w^3 \) terms is 16, so those like terms together become \( 16w^3 \). This process simplifies the expression.
Polynomial Simplification
Simplifying a polynomial means reducing it to its simplest form. After combining the coefficients of like terms, rewrite the simplified expression by grouping all the results together. In our example, after combining the like terms \( 3w^3, 5w^3, -4w^3, 12w^3 \) to get \( 16w^3 \), and noting that \( -2w^2 \) didn't have any similar terms to combine with, the final step is to put it all together. Thus, the simplified polynomial expression becomes:

\( 16w^3 - 2w^2 \). This new expression is easier to understand and work with because it no longer has redundant terms. Always remember that the goal is to make the expression as concise as possible while retaining the same 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

Write an algebraic expression for each problem. Parthenon. To obtain a pleasing rectangular shape, the ancient Greeks constructed buildings with a length that was about \(\frac{1}{6}\) longer than the width. If the width of the Parthenon is \(x\) meters and its length is \(x+\frac{1}{6} x\) meters, then what is the perimeter? What is the area?

Combine like terms. $$ 9 m n-m n $$

Find each product or quotient. $$ 3\left(\frac{y}{3}\right) $$

Exploration Find the decimal representations of $$ \frac{2}{9}, \frac{2}{99}, \frac{23}{99}, \frac{23}{999}, \frac{234}{999}, \frac{23}{9999}, \text { and } \frac{1234}{9999} $$ a) What do these decimals have in common? b) What is the relationship between each fraction and its decimal representation?

Perform these computations. $$-29-0.3$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied 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.