Problem 29 - Elementary and Intermediate Algebra

Question

Find each product. See Example 2. $$ (3 n+1)(3 n-1) $$

Step-by-Step Solution

Answer

The product is \(9n^2 - 1\).

Step 1: Identify the Binomial Form

The expression \((3n + 1)(3n - 1)\) is a product of two binomials. This is of the form \((a + b)(a - b)\), known as the difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is given by \((a + b)(a - b) = a^2 - b^2\). Here, we identify \(a = 3n\) and \(b = 1\). Substitute these into the formula.

Step 3: Calculate \(a^2\) and \(b^2\)

Calculate each square:\[a^2 = (3n)^2 = 9n^2\]\[b^2 = 1^2 = 1\]

Step 4: Substitute Back into the Formula

Substitute the calculated squares into the difference of squares formula:\[a^2 - b^2 = 9n^2 - 1\]

Step 5: Simplify the Expression

The expression \(9n^2 - 1\) is already simplified, representing the product of \((3n + 1)(3n - 1)\).

Key Concepts

Binomials

A binomial is an algebraic expression that consists of exactly two terms. Each of these terms might include numbers, variables, or both. Binomials play a critical role in the world of algebra, as they are the building blocks for constructing equations and expressions. When working with binomials, you often encounter operations like addition, subtraction, and multiplication. In our example

• The terms "3n + 1" and "3n - 1" are both binomials.
• Binomials often make problems interesting due to their simplicity and structure.
• Understanding how to manipulate binomials forms the foundation for mastering more complex algebraic operations.

The binomial formula \((a + b)(a - b) = a^2 - b^2\) exploits the potential of these expressions by transforming them into a product of a simpler entity. This simplification is crucial when solving algebra equations or simplifying polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They are used to express mathematical relationships and can range from simple expressions to more intricate ones. It is vital to learn how to manipulate algebraic expressions effectively, as they are foundational in problem-solving across various levels of mathematics.In the example given:

• The expression \((3n + 1)(3n - 1)\) is an algebraic expression involving multiplication of two binomials.
• Within this expression, "3n" is a term that consists of a coefficient (3) and a variable (n), while "1" is a constant term.
• Using algebraic expressions allows us to handle unknowns and solve for variables effectively.

Understanding how to work with algebraic expressions, including recognizing forms like the difference of squares, is crucial for simplifying equations and finding solutions efficiently.

Multiplication of Polynomials

When multiplying polynomials, it's important to apply specific algebra rules that make the process systematic and logical. The most basic polynomials in this context are binomials, like those found in the example. The multiplication of binomials is a critical skill, often using the FOIL method or special formulas such as the difference of squares.

• The rule for the difference of squares is incredibly useful: \((a + b)(a - b) = a^2 - b^2\).
• This formula simplifies the multiplication of two binomials into a simpler expression: in this case, \(9n^2 - 1\).
• Seeing these polynomials multiplied and reduced into simpler terms provides insight into how complex expressions can be manageable and understandable.

By mastering multiplication of polynomials, you also become equipped to handle equations that involve multiple terms, setting a strong foundation for tackling more advanced algebraic challenges.