Problem 79 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(x-7)(x+7)$$

Step-by-Step Solution

Answer

\(x^2 - 49\)

Step 1: Identify the Expression

We are given the expression \((x-7)(x+7)\). This is a special product known as the difference of squares.

Step 2: Apply the Difference of Squares Formula

Recall that the difference of squares formula is \((a-b)(a+b) = a^2 - b^2\). In this case, identify \(a = x\) and \(b = 7\).

Step 3: Substitute and Simplify

Substitute \(a = x\) and \(b = 7\) into the formula, giving us \(x^2 - 7^2\).

Step 4: Calculate the Square

Calculate the square of 7, which is \(7^2 = 49\).

Step 5: Write the Final Answer

The expression simplifies to \(x^2 - 49\).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic pattern that makes multiplying certain mathematical expressions easier and faster. This pattern occurs when you have two terms, let's call them "a" and "b," being subtracted and added, like

• \((a-b)(a+b)\).

When expanded, the product results in:

• \(a^2 - b^2\).

Why is this important? Recognizing patterns such as the difference of squares allows you to simplify the multiplication process without having to multiply each term individually as normally required in binomial expansion. In the example

• \((x-7)(x+7)\),

"a" is \(x\) and "b" is \(7\), simplifying directly to

• \(x^2 - 49\).

This formula is handy because it turns a potentially tedious task into something straightforward and fast, allowing you to solve algebra problems more efficiently.

Mathematical Expressions

In algebra, mathematical expressions are combinations of numbers, variables, and arithmetic operations that represent a value. Each component of an expression plays a particular role:

• **Variables**: Symbols like \(x\) or \(y\) that represent unknown values or values that can change.
• **Constants**: Fixed values like \(7\) in our exercise.
• **Operators**: Symbols like "+" and "-" that indicate operations performed on variables and numbers.

The goal when working with expressions is often to simplify or solve them, finding the value of the variables they contain or transforming them into simpler, equivalent forms. In algebra, recognizing structures and patterns, such as the difference of squares, aids in simplifying mathematical expressions efficiently. Being able to identify and manipulate these expressions is key to mastering algebra and tackling more complex problems later on. Grasping how to work with them ensures success in both basic and advanced mathematical contexts.

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials together to produce a new polynomial. This process generally requires distributing each term in one polynomial to every term in the other. In our example of

• \((x-7)(x+7)\),

it specifically uses the difference of squares, which simplifies the multiplication process dramatically. Ordinarily, you'd multiply each term by each term, but here you use the formula without expanding traditionally. Here’s a breakdown of a standard multiplication solution without the pattern:

• Multiply \(x\) by \(x\) to get \(x^2\).
• Multiply \(x\) by \(7\) to get \(7x\).
• Multiply \(-7\) by \(x\) to get \(-7x\)."
• Multiply \(-7\) by \(7\) to get \(-49\).

Notice how the middle terms, \(7x\) and \(-7x\), cancel out:

• \(x^2 - 49\).

Using such patterns as the difference of squares not only simplifies this step but also reduces potential errors from manual multiplication. Understanding and mastering these kinds of shortcuts is an essential part of gaining proficiency in algebra.