Problem 5 - Basic Technical Mathematics with Calculus

Question

Find the indicated products directly by inspection. It should not be necessary to write down intermediate steps [except possibly when using Eq. (6.6) ] $$40(x-y)$$

Step-by-Step Solution

Answer

The product is \(40x - 40y\).

Step 1: Identify the Components

In the expression \(40(x-y)\), the components are the constant 40 and the binomial \((x-y)\).

Step 2: Apply Distribution

Recognize that the number 40 needs to be distributed across the binomial \((x-y)\). This means multiplying 40 by each term within the parentheses separately.

Step 3: Multiply the First Term

Multiply 40 by \(x\): \[40 \times x = 40x\]

Step 4: Multiply the Second Term

Multiply 40 by \(-y\): \[40 \times (-y) = -40y\]

Step 5: Combine the Results

Combine the results from Steps 3 and 4 to form the final expression:\[40x - 40y\]

Key Concepts

Understanding the Distribution Property

The distribution property is a fundamental principle in algebra that helps simplify expressions and solve equations. It allows you to multiply a sum or difference by distributing a factor to each term inside the parenthesis.

When you see an expression like \(40(x-y)\), the distribution property tells us to multiply the 40 by each term in the binomial \((x-y)\). Here’s how it works:

• Multiply 40 by \(x\) to get \(40x\).
• Multiply 40 by \(-y\) to get \(-40y\).

After applying distribution, the expression becomes \(40x - 40y\). It transforms terms inside parentheses into a simpler form. This skill is useful for working through more complicated algebraic problems efficiently.

Exploring Binomials

A binomial is a simple and common algebraic expression consisting of two terms separated by a plus or minus sign. For example, in our exercise, \((x-y)\) is a binomial because it consists of the terms \(x\) and \(-y\).

Binomials are everywhere in algebra, and understanding how to handle them is crucial. They often appear in equations, polynomial expressions, and may even need to be manipulated or expanded. Learning to work with binomials involves recognizing their structure:

• Look for two distinct terms joined by a plus or minus.
• Understand that distribution or other operations can be used on each term separately.

In the exercise, the binomial \((x-y)\) was a crucial part, and recognizing it allowed us to correctly apply the distribution property.

Mastering Multiplication of Terms

Multiplying terms involves combining numbers and variables through a multiplication operation. When dealing with expressions like \(40(x-y)\), it's important to multiply carefully to maintain accuracy.

Here's the process broken down:

• Multiply a number by a variable (e.g., 40 multiplied by \(x\) becomes \(40x\)).
• Don't forget the sign of each term; multiplying 40 by \(-y\) gives \(-40y\), not \(40y\).

Simple multiplication rules for signs also apply:

• A positive times a negative gives a negative.
• Positive times positive and negative times negative both result in a positive.

Multiplying terms accurately ensures your final expression is correct, just like converting \(40(x-y)\) into \(40x - 40y\).