Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. $$ (2 u-v)(2 u+v) $$

Step-by-Step Solution

Answer

The simplified expression after performing the indicated operations and simplifying is \(4u^2 - v^2\).

Step 1: Identify the problem

We are given a multiplication problem with two binomials: \[ (2u - v)(2u + v) \]

Step 2: Apply the distributive property

To perform the multiplication, we will use the distributive property (also known as FOIL): \[ (2u - v)(2u + v) = (2u)(2u) + (2u)(v) - (v)(2u) - (v)(v) \]

Step 3: Simplify the terms

Now, we simplify the terms of the expression from Step 2: \[ 4u^2 + 2uv - 2uv - v^2 \]

Step 4: Combine like terms

Now, we will combine like terms from the expression in Step 3: \[ 4u^2 + 2uv - 2uv - v^2 = 4u^2 - v^2 \]

Step 5: Write the final answer

The simplified expression is: \[ 4u^2 - v^2 \]

Key Concepts

The Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term across the terms within a parenthesis. This is particularly useful when dealing with binomial multiplication.

For example, when given \[(a + b)(c + d)\], we use the distributive property to multiply each term in the first parenthesis by each term in the second parenthesis:

• First, multiply \(a\) with \(c\) and \(a\) with \(d\).
• Next, multiply \(b\) with \(c\) and \(b\) with \(d\).

This operation helps expand the expression into a form which can be further simplified or solved.In our problem with \[(2u - v)(2u + v)\], applying the distributive property gives us:\[(2u)(2u) + (2u)(v) - (v)(2u) - (v)(v)\].By clearly understanding and applying this property, you can break down complex algebraic expressions into simpler parts.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as simple as possible. Once we have expanded an expression using the distributive property, our next step is to simplify that expression.

In simpler terms, you replace longer expressions with equivalent shorter ones by performing operations and reducing where possible. For instance, after applying the distributive property in our example, we have:\[4u^2 + 2uv - 2uv - v^2\].Each term is simplified by carrying out basic arithmetic operations:

• Multiplying coefficients and combining products, like \((2u)(2u) = 4u^2\).
• Performing the same for other parts of the expression.

Simplifying helps you see patterns or opportunities to combine terms later on.

Combining Like Terms

Combining like terms is a key step in simplifying expressions. "Like terms" are terms whose variables (and their exponents) match, although their coefficients may differ. For example, in the expression: \[3x + 2x\],\(3x\) and \(2x\) are like terms because both are multiples of \(x\). The process involves adding or subtracting these coefficients while keeping the variable part unchanged. In our example, \[4u^2 + 2uv - 2uv - v^2\],\(2uv\) and \(-2uv\) are like terms. Combining them gives us:

• \(2uv - 2uv = 0\), simplifying the expression further to.

\[4u^2 - v^2\].Combining like terms effectively reduces complexity, ultimately presenting the expression in its most straightforward form.