Problem 84 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(x+2 y)(x-2 y)$$

Step-by-Step Solution

Answer

The product is \(x^2 - 4y^2\).

Step 1: Recognize the Pattern

The expression is in the form \((a + b)(a - b)\) where \(a = x\) and \(b = 2y\). This is a difference of squares pattern.

Step 2: Apply the Difference of Squares Formula

Using the identity \((a + b)(a - b) = a^2 - b^2\), substitute \(a = x\) and \(b = 2y\) into the expression to get \(x^2 - (2y)^2\).

Step 3: Simplify the Squares

Calculate the squares: \(x^2\) remains as is, and \((2y)^2 = 4y^2\). Now, the expression is \(x^2 - 4y^2\).

Key Concepts

Difference of Squares

The difference of squares is a useful algebraic identity that simplifies expressions of the form \((a+b)(a-b)\). When you see an expression like this, it's an indication that you can use a shortcut formula to simplify it. The formula is:

• \((a+b)(a-b) = a^2 - b^2\)

This means that you simply need to square each term in the binomials, \(a\) and \(b\), and then subtract the square of the second term from the square of the first term.
In the exercise given, \(a\) corresponds to \(x\) and \(b\) corresponds to \(2y\). Hence, applying the difference of squares formula simplifies \((x+2y)(x-2y)\) directly to \(x^2 - (2y)^2\). Recognizing this pattern allows for quick simplification without having to multiply out each term individually.

Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by each term in the other polynomial. When dealing with binomials, a common and efficient method used is the FOIL method, representing:

• First
• Outer
• Inner
• Last

For \((x + 2y)(x - 2y)\):

• First: \(x \cdot x = x^2\)
• Outer: \(x \cdot (-2y) = -2xy\)
• Inner: \(2y \cdot x = 2xy\)
• Last: \(2y \cdot (-2y) = -4y^2\)

Adding these results together, you get \(x^2 - 2xy + 2xy - 4y^2\). Notice that the middle terms \(-2xy\) and \(+2xy\) cancel each other out, confirming that the expression simplifies to \(x^2 - 4y^2\), consistent with the difference of squares formula. Understanding how to apply this method ensures that you multiply polynomials accurately every time.

Simplification of Expressions

Simplification of expressions is about reducing an algebraic expression to its most compact or simplest form. It's a crucial step in algebra that makes working with expressions much easier. In this context, recognize and simplify common patterns like the difference of squares, where unnecessary terms can be eliminated.
In the exercise, once we've applied the difference of squares formula and calculated the squares, we are left with \(x^2 - 4y^2\). The goal was to simplify \((x+2y)(x-2y)\) through techniques like the difference of squares which allowed us to bypass additional steps.

• A simplified expression is easier to evaluate and communicate.
• Simplifying can reveal further insights or properties about the expression.

Through simplification, you're not changing the value, just the way the expression looks, making it more manageable and efficient for further mathematical operations.