Problem 81 - Algebra A Combined Function

Question

Multiply. See Section 5.6. \((x-4)(x+4)\)

Step-by-Step Solution

Answer

The result is \( x^2 - 16 \).

Step 1: Recognize the Pattern

Notice that the given expression \( (x-4)(x+4) \) follows the pattern of a difference of squares, which is \((a-b)(a+b) = a^2 - b^2\).

Step 2: Identify a and b Values

For the expression \( (x-4)(x+4) \), identify \a\ as \x\ and \b\ as \4\.

Step 3: Apply the Difference of Squares Formula

Substitute \a = x\ and \b = 4\ into the difference of squares formula: \((x-4)(x+4) = x^2 - 4^2\).

Step 4: Simplify the Expression

Calculate \4^2\, which is \16\. Thus, the expression becomes \[ x^2 - 16 \].

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They are used extensively in algebra to represent mathematical problems in a general form. An algebraic expression can range from very simple, like \(x + 2\), to more complex scenarios involving multiple operations and variables.

In the provided exercise, \((x-4)(x+4)\) is an algebraic expression illustrating the difference of squares. Recognizing such patterns can significantly simplify complex algebraic manipulations.

Key features of algebraic expressions include:

• Variables, which are symbols like \(x\) or \(y\) that stand in for unknown numbers.
• Constants are fixed numbers, such as the numbers \(-4\) and \(4\) in the given expression.
• Operators, which are mathematical symbols like \(+\), \(-\), \(\times\), and \(\div\).

Understanding how to manipulate these expressions is crucial for solving algebraic equations and understanding mathematical relationships.

Multiplying Binomials

Multiplying binomials involves applying the distributive property to expand expressions that contain two terms within brackets. It is a critical skill in algebra that allows you to simplify and solve equations. A standard method to multiply binomials is the FOIL method, which stands for First, Outer, Inner, and Last. However, there are faster methods like spotting specific patterns such as the difference of squares.

In the expression \((x-4)(x+4)\), recognizing it as a difference of squares allows for a quicker multiplication process:

• First: Multiply the first terms, \(x \times x = x^2\).
• Outer: Multiply the outer terms, \(x \times 4 = 4x\).
• Inner: Multiply the inner terms, \(-4 \times x = -4x\).
• Last: Multiply the last terms, \(-4 \times 4 = -16\).

Notice that the middle terms \(4x\) and \(-4x\) cancel each other out, resulting in the simplified form \(x^2 - 16\). Recognizing the difference of squares is key as it bypasses the need to use all four steps.

Factoring Techniques

Factoring involves reversing the process of multiplication to express an equation or expression as a product of its factors. In algebra, factoring is essential for solving polynomial equations and simplifying expressions.

The difference of squares is one specific technique where a binomial expression such as \(a^2 - b^2\) is factored into \((a-b)(a+b)\). Recognizing this pattern can simplify solving algebraic problems without needing lengthy expansion processes.

Here are the general steps to apply difference of squares factoring:

• Identify expressions in the form of \(a^2 - b^2\).
• Express the terms squared, as seen in \(x^2 - 16\), where \(x\) is \(a\) and \(4\) is \(b\).
• Factor the expression as \((x-4)(x+4)\).

By mastering these techniques, the complexity of solving polynomials or algebraic expressions decreases significantly. Understanding these fundamental concepts aids in deeper mathematical problem-solving.