Problem 73 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (n+6)(n-6) $$

Step-by-Step Solution

Answer

The result is n^2 - 36.

Step 1: Identify the Expression Type

The expression (n+6)(n-6) is a difference of squares, which is a special product pattern. It matches the form (a+b)(a-b) where we can identify a = n and b = 6.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is (a+b)(a-b) = a^2 - b^2. Using a = n and b = 6, the expression becomes (n+6)(n-6) = n^2 - 6^2.

Step 3: Simplify the Expression

Now, simplify 6^2 to get 36. Therefore, the expression becomes n^2 - 36.

Key Concepts

Difference of Squares

The term "difference of squares" might sound complicated, but it's actually quite simple. It involves a specific pattern in algebraic expressions. This pattern happens when you have two binomials that look like this:

• \((a+b)(a-b)\)

"Difference" refers to subtraction, and "squares" refers to the squares of terms. So, this pattern is about subtracting one square number from another. For example, a square number is one like \(a^2\).

The basic formula goes

• \((a+b)(a-b) = a^2 - b^2\)

Once you recognize this pattern, simplifying the expression becomes easy. With practice, identifying terms \(a\) and \(b\) becomes second nature, making these problems almost routine.

Special Product Patterns

Special product patterns are recurring, easy-to-recognize patterns in math that allow you to simplify expressions more efficiently. These patterns work like shortcuts. Knowing them can save you time and effort. The difference of squares is one such pattern, but there are others, like:

• Perfect square trinomials: \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\)
• Sum and difference of cubes

Recognizing where these patterns fit helps in factoring and expanding polynomial expressions quickly. These concepts underpin a lot of algebraic manipulation, and being familiar with them is essential for mastering algebra.

Polynomial Simplification

Polynomial simplification is about breaking down complex expressions into simpler forms. Think of it like untangling a knot of variables and numbers. Simplifying involves combining like terms, factoring, or using formulas like the difference of squares to make expressions more manageable.

When you simplify, keep these tips in mind:

• Identify and combine like terms, such as \(3x\) and \(5x\)
• Apply known formulas or patterns to recognize opportunities for simplification
• Break down complex expressions into their simplest terms: e.g., \((n+6)(n-6)\) simplifies to \(n^2 - 36\)

The goal is to reach a form that is easy to interpret or further manipulate, making the expression more understandable and usable.