Problem 60 \(\left(a^{3}-2 a^{2} b\right)+\... [FREE SOLUTION]

Chapter 5: Problem 60

\(\left(a^{3}-2 a^{2} b\right)+\left(a b^{2}+b^{3}\right)-\left(3 a^{2} b+4 a b^{2}\right)\)

Short Answer

Expert verified

a^3 - 5a^2b - 3ab^2 + b^3

Step by step solution

- Distribute the Minus Sign

Rewrite the expression by distributing the minus sign through the parentheses. This changes the expression to: a^3 - 2a^2b + ab^2 + b^3 - 3a^2b - 4ab^2

- Combine Like Terms

Identify and combine like terms:- Combine the terms with \(a^3\): \(a^3\)- Combine the terms with \(a^2b\): -2a^2b - 3a^2b = -5a^2b- Combine the terms with \(ab^2\): ab^2 - 4ab^2 = -3ab^2- Combine the terms with \(b^3\): b^3The resulting expression is: a^3 - 5a^2b - 3ab^2 + b^3

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.

Distributing Minus Sign

When simplifying polynomial expressions, it's crucial to handle negative signs correctly. This is often called 'distributing the minus sign.' To simplify the given polynomial, we distribute the minus sign to each term inside the parentheses.

For example, if you have \(- (3a^2 b + 4ab^2)\), you distribute the minus as follows: \- 3a^2 b - 4ab^2\. This changes each term's sign from positive to negative.

Distributing the minus sign helps to remove parentheses and make combining like terms easier in the next steps. Always be vigilant when you distribute a minus sign to ensure all signs within the parentheses are correctly changed.

Combining Like Terms

After distributing the minus sign, the next step in polynomial simplification is combining like terms. Like terms in polynomials are terms that have the same variables raised to the same powers.

In the expression \(a^3 - 2a^2b + ab^2 + b^3 - 3a^2b - 4ab^2\), we have:

\text{\textbackslash(left a\right\brace ^\textbackslash3\textbrace\right\textbrace }:
- Combine \(-2a^2b - 3a^2b\) to get \-5a^2b\.
- Combine \(ab^2 - 4ab^2\) to get \-3ab^2\.
\text{\textbackslash(left b^\textbackslash3\right):

The resulting simplified expression is \(a^3 - 5a^2b - 3ab^2 + b^3\). Combining like terms simplifies the polynomial down to its simplest form.

Polynomial Expressions

Polynomials are mathematical expressions that consist of variables, coefficients, and exponents. Understanding polynomial expressions is fundamental in algebra.

A polynomial can have several terms and is formed like this: \(ax^m + bx^(m-1) + ... + k\), where \(a, b, ... , k\) are coefficients and \(m, m-1, ... , 0\) are non-negative integers.

To simplify a polynomial expression:

First, distribute any minus signs or factors inside parentheses.
Then, identify and combine like terms to reduce the polynomial to its simplest form.

Mastering polynomials and their simplification is essential for progressing in algebra and tackling more complex mathematical problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

\(\left(a^{3}-2 a^{2} b\right)+\left(a b^{2}+b^{3}\right)-\left(3 a^{2} b+4 a b^{2}\right)\)

Short Answer

Step by step solution

- Distribute the Minus Sign

- Combine Like Terms

Key Concepts

Distributing Minus Sign

Combining Like Terms

Polynomial Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Pure Maths

Probability and Statistics

Logic and Functions

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.