LoginSign up
Log outGo to App
Go to App
Learning Materials
- Explanations
Explanations
- Biology
- Business Studies
- Chemistry
- Chinese
- Combined Science
- Computer Science
- Economics
- Engineering
- English
- English Literature
- Environmental Science
- French
- Geography
- German
- History
- Human Geography
- Italian
- Law
- Macroeconomics
- Marketing
- Math
- Microeconomics
- Nursing
- Physics
- Politics
- Psychology
- Sociology
- Spanish
- Textbooks
Textbooks
- Biology
- Business Studies
- Chemistry
- Combined Science
- Computer Science
- Economics
- English
- Environmental Science
- Geography
- History
- Math
- Physics
- Psychology
- Sociology
Features
- Flashcards
- Vaia IA
- Notes
- Study Plans
- Study Sets
- Exams
Discover
- Magazine
- Find a job
- Mobile App
Chapter 0: Problem 37
Multiply and simplify. $$ (3 y+6)\left(\frac{1}{3} y^{2}-5 y-4\right) $$
Short Answer
Expert verified
The simplified product is \(y^3 - 13y^2 - 42y - 24\).
Step by step solution
01
- Distribute each term
Distribute each term in the first binomial, \(3y + 6\), across each term in the second polynomial, \left( \frac{1}{3} y^2 - 5y - 4 \right)\.
02
- Multiply 3y with each term
Multiply \(3y\) by each of the three terms in the second polynomial: \(3y \times \frac{1}{3} y^2 \), \( 3y \times -5y \), and \(3y \times -4\).
03
- Simplify the multiplications
Perform each multiplication: \(3y \times \frac{1}{3} y^2 = y^3\), \(3y \times -5y = -15y^2\), and \(3y \times -4 = -12y\).
04
- Multiply 6 with each term
Multiply \(6\) by each of the three terms in the second polynomial: \(6 \times \frac{1}{3} y^2 \), \( 6 \times -5y \), and \(6 \times -4\).
05
- Simplify the multiplications
Perform each multiplication: \(6 \times \frac{1}{3} y^2 = 2y^2\), \(6 \times -5y = -30y\), and \(6 \times -4 = -24\).
06
- Combine all terms
Combine all the terms obtained from the distribution: \(y^3 - 15y^2 - 12y + 2y^2 - 30y - 24 \).
07
- Simplify the polynomial
Combine like terms: Combine the \y^2\ and \y\ terms: \(y^3 - 13y^2 - 42y - 24\).
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Distribution
When faced with multiplying binomials and polynomials, understanding the concept of binomial distribution is key. This involves distributing each term of the binomial across each term of the polynomial. In our example, we started with \(3y + 6\) and distributed it to each term in the polynomial \[ \frac{1}{3} y^2 - 5y - 4 \].
This step-by-step approach allows us to systematically handle the multiplication and ensures we're considering each combination of terms, thereby avoiding errors. Remember to keep track of both numerical coefficients and variable terms to ensure accuracy. Breaking down the problem helps in managing the distribution process efficiently.
Combining Like Terms
Once all terms are distributed and multiplied, we'll need to combine like terms. These are terms that have the same variables raised to the same power. For instance, in our example after distributing we get: \[ y^3 - 15y^2 - 12y + 2y^2 - 30y - 24\].
Notice the \( y^2\) terms \( -15y^2\) and \(2y^2\) are like terms, as well as the \( y\) terms \( -12y\) and \( -30y\).
Combining them, we get: \[ y^3 - 13y^2 - 42y - 24 \]. \ Properly combining like terms simplifies the polynomial and clarifies the final result. This concept is crucial for cleaning up our answers and achieving the simplest form.
Polynomial Simplification
Polynomial simplification entails reducing the polynomial expressions to their simplest form. After we have combined our like terms, our goal is to make the expression as concise as possible. This involves eliminating any unnecessary terms and ensuring all like terms are combined.
In our example, we went from \[y^3 - 15y^2 - 12y + 2y^2 - 30y - 24 \] to \[ y^3 - 13y^2 - 42y - 24 \].
The simplified polynomial is easier to read, understand, and work with for further algebraic operations.
Always remember to carefully review each step during simplification to ensure no terms are overlooked.
Algebraic Expressions
Algebraic expressions consist of variables, constants, and arithmetic operations. In our problem, we are dealing with expressions involving multiple terms. Each term is a component of the larger polynomial, such as \[ 3y, 6, \frac{1}{3}y^2, -5y, \text{and}-4\].
Understanding how to manipulate these expressions through addition, subtraction, and multiplication is fundamental in algebra.
In the given exercise, multiplying binomials by higher-degree polynomials exemplifies how flexible and dynamic algebraic expressions can be. Knowing how to handle these expressions through correct distribution and combination enables us to solve more complex problems and enhances our algebra skills significantly. Always approach your expressions methodically to ensure clarity and correctness.
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
Recommended explanations on Math Textbooks
Logic and Functions
Read ExplanationGeometry
Read ExplanationMechanics Maths
Read ExplanationTheoretical and Mathematical Physics
Read ExplanationDiscrete Mathematics
Read ExplanationApplied Mathematics
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.