Problem 234 - Elementary Algebra

Question

Factor. \(n^{2}-9\)

Step-by-Step Solution

Answer

\(n^{2} - 9\) factors to \((n + 3)(n - 3)\).

Step 1 - Identify the Form

Recognize that the given expression is a difference of squares. It can be written in the form \[a^{2} - b^{2}\]. For the given expression, \(n^{2} - 9\), identify \(a\) and \(b\).

Step 2 - Write as a Difference of Squares

Express \(n^{2} - 9\) in the form \(a^{2} - b^{2}\). Here, \(a = n\) and \(b = 3\) because \(3^{2} = 9\). Hence, \(n^{2} - 9\) can be written as \((n^{2} - 3^{2})\).

Step 3 - Apply the Difference of Squares Formula

Use the difference of squares formula \(a^{2} - b^{2} = (a + b)(a - b)\). Substituting \(a = n\) and \(b = 3\), you get:\[^{2} - 9 = (n + 3)(n - 3)\].

Step 4 - Write the Final Factorization

Therefore, the factorization of \(n^{2} - 9\) is \((n + 3)(n - 3)\).

Key Concepts

Factoring

Factoring is an essential skill in algebra that refers to breaking down an expression into simpler elements, called factors, that, when multiplied together, give the original expression. Imagine factoring like taking a big item and breaking it down into smaller pieces.

For example, in the expression \( n^{2} - 9 \), we recognized that it was a difference of squares, which is a special form in factoring. Recognizing patterns like this makes factoring much easier.

Let's break it down:

• We noticed that \( n^{2} - 9 \) follows the pattern \( a^{2} - b^{2} \).
• This tells us we can factor it into two smaller binomials.
• Using the difference of squares formula \(a^{2} - b^{2} = (a + b)(a - b)\), we factored \( n^{2} - 9 \) into \( (n + 3)(n - 3) \).

By mastering factoring, you can simplify complex problems and find solutions faster.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( n \), \( x \), \( y \)), and arithmetic operations (like addition, subtraction, multiplication, and division). These expressions represent specific values based on the variables involved.

In the problem we tackled, our algebraic expression was \( n^{2} - 9 \). This showed us the power and simplicity of variables and constants in describing numbers and operations:

• \( n^{2} \) is the variable part where the value of \( n \) can change.
• \( 9 \) is a constant, a fixed number.
• When combined, they create an algebraic expression that we can work with.

Algebraic expressions are foundational in mathematics, offering a way to generalize and solve problems efficiently.

Polynomial Factorization

Polynomial factorization is an important process where we write a polynomial as a product of its factors. A polynomial is an algebraic expression with more than one term and can be factored in various ways, depending on its structure.

In our example with \( n^{2} - 9 \), we specifically looked at a polynomial that fits the 'difference of squares' method:

• First, we identified it in the form of \( a^{2} - b^{2} \).
• Then, we used the difference of squares formula to break it down into \( (n + 3)(n - 3) \).

This approach helps simplify polynomials and solve equations. Learning different factorization methods, such as grouping and using special formulas, will make handling more complicated polynomials easier. By understanding and practicing polynomial factorization, you prepare yourself for more advanced algebra and calculus concepts.