Problem 42 Multiply. $$a b\left(2 a^{2}-4 a... [FREE SOLUTION]

Chapter 9: Problem 42

Multiply. $$a b\left(2 a^{2}-4 a b-6 b^{2}\right)$$

Short Answer

Expert verified

The result of the multiplication is $ 2a^3b - 4a^2b^2 - 6ab^3 $

Step by step solution

Distribute a*b to each term inside the parenthesis

Begin by multiplying each term of the polynomial $2a^2 - 4ab - 6b^2$ by $ab$, which gives: $ab * 2a^2 - ab * 4ab - ab * 6b^2$

Simplify each term

Now, simplify each term by calculating the multiplications and consolidate like terms: $ 2a^3b - 4a^2b^2 - 6ab^3 $

Write the final answer in descending order of powers

The order of terms is not important in the answer but it's customary to write it in descending order of powers so the result would be: $ 2a^3b - 4a^2b^2 - 6ab^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.

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by all the terms within a parenthesis. In our example, we are using the distributive property to multiply the term $ ab $ with each of the terms inside the parenthesis $ 2a^2 - 4ab - 6b^2 $.

To do this, we perform the following actions:

Multiply $ ab $ by $ 2a^2 $ to get $ ab \times 2a^2 $.
Multiply $ ab $ by $ -4ab $ to get $ ab \times -4ab $.
Multiply $ ab $ by $ -6b^2 $ to get $ ab \times -6b^2 $.

By applying the distributive property, we ensure that all terms inside the parentheses are individually multiplied by the external term, leading to a fully expanded expression.

Simplifying Expressions

When simplifying expressions, our goal is to rewrite them in a simpler or more concise form while maintaining their original meaning. In mathematical expressions, this involves performing arithmetic operations and combining like terms.

In our exercise, we simplify each term resulting from the distribution:

For $ ab \times 2a^2 $, we combine powers of $ a $, resulting in $ 2a^3b $.
For $ ab \times -4ab $, we combine like terms, giving us $ -4a^2b^2 $.
For $ ab \times -6b^2 $, we end up with $ -6ab^3 $.

Notice how we organize these terms by their power of $ a $ and $ b $ to create a neat and standard expression, simplifying communication and calculation.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They represent general mathematical relationships and can be manipulated to solve problems.

In our given problem, the expression $ ab(2a^2 - 4ab - 6b^2) $ makes use of variable terms, multipliers, and operations to express a relationship. Terms within the expression can have:

Variables, such as $ a $ and $ b $. These represent unknown or general quantities.
Coefficients, such as 2, -4, and -6. These are numbers multiplying the variables.
Operators, such as addition and subtraction, that join different terms together.

Understanding algebraic expressions is crucial in mathematics as they form the basis for equations and further algebraic operations, acting like the language through which mathematical reasoning is conveyed.

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

Multiply. $$a b\left(2 a^{2}-4 a b-6 b^{2}\right)$$

Short Answer

Step by step solution

Distribute a*b to each term inside the parenthesis

Simplify each term

Write the final answer in descending order of powers

Key Concepts

Distributive Property

Simplifying Expressions

Algebraic Expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Probability and Statistics

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.