Problem 33 - College Algebra

Question

Find each product. $$(3 x+2)(3 x-2)$$

Step-by-Step Solution

Answer

The product of \((3x+2)(3x-2)\) is \(9x^2 - 4\).

Step 1: Identify the terms

Identify terms that can be grouped based on the concept of difference of two squares. Here, the terms are \(a = 3x\) and \(b = 2\).

Step 2: Apply the formula for the difference of two squares

Apply the formula \(a^2 - b^2 = (a+b)(a-b)\). Multiply the given expression accordingly: \((3x + 2)(3x - 2)\). Here, \(a+b = 3x + 2\) and \(a-b = 3x - 2\)

Step 3: Compute the result

Perform the multiplication. When we multiply out the brackets we should get \(9x^2 - 4\) as the product.

Key Concepts

Polynomial Multiplication

When we talk about polynomial multiplication, we're discussing the process of multiplying two polynomials together. A polynomial is an algebraic expression consisting of variables, coefficients, and exponents that are combined using addition, subtraction, and multiplication.

Consider the polynomial \(3x + 2\). This is a binomial because it has two terms. When multiplying it by another binomial, say \(3x - 2\), we use certain methods to find the product. The result we're aiming for is another polynomial where each term of the first polynomial has been multiplied by each term of the second polynomial. The difference of two squares is a special case of polynomial multiplication where the result reduces to a simpler form.

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and arithmetic operations. Algebraic expressions are fundamental in algebra and serve as the building blocks for more complex equations. For instance, \(3x + 2\) and \(3x - 2\) from our example are algebraic expressions that include the variable \(x\), coefficients \(3\) and \(2\), and arithmetic operations (addition and subtraction).

Understanding these expressions allows students to perform operations such as polynomial multiplication effectively. Algebraic expressions can be combined, broken down, and transformed, providing a powerful tool for solving a wide range of mathematical problems.

Binomial Products

The term 'binomial products' refers to the result of multiplying two binomials together. A binomial is, as the name suggests, an algebraic expression with exactly two terms. When two binomials are multiplied, the result is a trinomial in most cases.

However, when dealing with a special case like the difference of two squares, we end up with a simpler product. This is what makes identifying such patterns crucial for simplifying polynomial multiplication. A binomial product like \(3x + 2\) times \(3x - 2\), after applying the formula for the difference of two squares, results in \(9x^2 - 4\), a two-term polynomial, rather than the usual three-term outcome.

FOIL Method

The FOIL method is a technique for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to a step-by-step process where each part of one binomial is multiplied by each part of the second binomial in a specific order.

For the given binomials \(3x + 2\) and \(3x - 2\), here's how we apply FOIL:

• First: Multiply the first terms in each binomial: \(3x \cdot 3x = 9x^2\).
• Outer: Multiply the outer terms: \(3x \cdot -2\).
• Inner: Multiply the inner terms: \(2 \cdot 3x\).
• Last: Multiply the last terms in each binomial: \(2 \cdot -2 = -4\).

The outer and inner terms cancel each other out in the case of the difference of two squares, hence why we don't see them in the final expression \(9x^2 - 4\). This simplification is a unique outcome of the specific pattern in our binomials.