Problem 51 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$(y-3) 6 y$$

Step-by-Step Solution

Answer

The expanded form is \(6y^2 - 18y\).

Step 1: Identify the Parts

The problem is to expand the expression \((y - 3) imes 6y\). Here, you have two parts: the binomial \((y - 3)\) and the monomial \(6y\). We will apply the distributive property to multiply these parts.

Step 2: Distribute the Monomial

Apply the distributive property by multiplying \(6y\) with each term inside the parenthesis: 1. Multiply \(6y\) by \(y\): the result is \(6y^2\).2. Multiply \(6y\) by \(-3\): the result is \(-18y\).

Step 3: Combine Results

Combine the results from Step 2:\[6y^2 - 18y\]. This is the expanded form of the original expression.

Key Concepts

Understanding Binomials

A binomial is a type of expression that contains exactly two terms. In our exercise, \((y-3)\) is the binomial because it consists of the two separate terms: \(y\) and \(-3\). Binomials can appear in different forms, often involving variables and constants.

• The first term, \(y\), is a variable. Variables are symbols that represent numbers and are used in expressions to generalize mathematics.
• The second term, \(-3\), is a constant, as it does not change its value.

Recognizing a binomial is essential in algebra as it helps you identify the structure of the expression and organize it accordingly.
This understanding is crucial as it allows you to apply mathematical properties, such as the distributive property, to perform operations effectively.

What is a Monomial

A monomial is an algebraic expression consisting of only one term. In the given problem, the monomial is \(6y\). This means it is a single mathematical term that may include numbers, variables, or the product of both.

• The number part of the term, called the coefficient, is \(6\) in \(6y\).
• The variable part is \(y\), which might represent different values depending on the context of the problem.

An important feature of monomials is that they do not have added or subtracted terms.
They can be as simple as a constant number or involve more complexity with variables raised to different powers, but they remain a single entity.

Expanding Expressions

Expanding an expression involves using mathematical properties to simplify or rewrite the expression. In this instance, we use the distributive property to expand \((y - 3) \times 6y\).
The distributive property tells us to multiply the single term outside the parenthesis by each term inside. Here’s a quick breakdown:

• The first part of the binomial, \(y\), is multiplied with the monomial \(6y\), resulting in \(6y^2\).
• The second part of the binomial, \(-3\), is multiplied with \(6y\), giving \(-18y\).

After distributing, you combine all these new terms together to form a simplified expression: \[6y^2 - 18y\].
By expanding expressions, you can more easily perform additional steps in problem-solving, like combining like terms or proceeding with solving equations.