/*! 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 16 Find each product. $$3 y\left(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 16

Find each product. $$3 y\left(y^{2}-2\right)$$

Short Answer

Expert verified
The product is \(3y^3 - 6y\).

Step by step solution

01

Apply the Distributive Property

To find the product, we'll use the distributive property, which states that for any numbers or expressions, \( a(b+c) = ab + ac \). Here, we'll distribute \(3y\) to each term inside the parentheses: \(3y(y^2 - 2) = 3y \cdot y^2 - 3y \cdot 2\).
02

Multiply Each Term

Now, we'll multiply \(3y\) with each term inside the parentheses separately. First, \(3y \cdot y^2 = 3y^3\) because \(y \cdot y^2 = y^{1+2} = y^3\). Then, multiply the second term: \(3y \cdot 2 = 6y\).
03

Write the Final Expression

Combine the results from the multiplication to get the final expression: \(3y^3 - 6y\).

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.

Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They represent values but do not have an equality sign like an equation. In the expression \(3y(y^2 - 2)\), we have components such as:
  • Variables: Symbols like \(y\) stand in place of numbers.
  • Constants: Specific numbers such as \(3\) and \(-2\) that are fixed values.
  • Operations: Actions like multiplication and subtraction that are performed on the variables and constants.
Understanding these elements is crucial for simplifying and manipulating expressions, such as performing multiplication using the distributive property.
Polynomial Multiplication
Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial. This is often done using the distributive property. Consider our expression \(3y(y^2 - 2)\). Here, \(3y\) is multiplied by each term inside the parentheses:
  • Distributive Property: We distribute the term outside the parentheses across every term inside. This results in \(3y \cdot y^2\) and \(3y \cdot (-2)\).
  • Simplification: Each of these products is simplified separately to give \(3y^3\) and \(-6y\).
After all terms have been multiplied, the results are combined to form the final expression. This approach is a fundamental technique for handling expressions that contain multiple terms.
Exponents
Exponents are a way to express repeated multiplication of a number by itself. For instance, in the term \(y^2\), \(y\) is multiplied by itself once. When multiplying expressions with the same base, you add the exponents together.
  • Multiplication Rule: In \(3y \cdot y^2\), you multiply the coefficients (\(3\) and the implied \(1\) of \(y^2\)), and add the exponents (\(y^1 \cdot y^2 = y^{1+2} = y^3\)).
  • Simplification: The multiplication results in \(3y^3\).
Understanding exponents is critical when working with algebra, as they frequently appear in polynomial expressions and require careful attention during multiplication and simplification.

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

Graph \(y=x^{2}\) and \(y=x^{3}\) in the first quadrant on the same coordinate plane. Explain which graph shows faster growth.

Find each sum or difference. $$(2 x-1)+5 x$$

Find each sum. $$\left(4 t-t^{2}\right)+(8 t+2)$$

Rewrite each expression using parentheses so that the terms having variables of the same power are grouped together. $$(a+2 b)+(3 a+b)$$

Evaluate each expression if \(a=-2, b=-6,\) and \(c=14\) $$a-b$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

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.