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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.

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