Problem 76 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The product is \(x^2 - 9y^2\).

Step 1: Understand the Problem

You need to find the product of two binomials: \((x - 3y)\) and \((x + 3y)\). This is an application of the difference of squares formula.

Step 2: Apply the Difference of Squares Formula

Recall the difference of squares formula: \((a-b)(a+b) = a^2 - b^2\). Identify \(a = x\) and \(b = 3y\) in the given expression \((x - 3y)(x + 3y)\).

Step 3: Substitute into the Formula

Substitute \(a = x\) and \(b = 3y\) into the difference of squares formula to get the equation: \(x^2 - (3y)^2\).

Step 4: Simplify the Expression

Simplify the equation \(x^2 - (3y)^2\). Compute \((3y)^2 = 9y^2\), leading to the simplified form: \(x^2 - 9y^2\).

Key Concepts

Binomial Products

When we talk about binomial products, we're dealing with the multiplication of two binomial expressions. A binomial expression is simply an algebraic expression that contains exactly two terms. For example, in the exercise
this would be \((x - 3y)\) and \((x + 3y)\). The process of multiplying these terms involves specific algebraic rules that simplify the multiplication process.

• First, you apply the distributive property which helps in expanding algebra. Essentially, this means you multiply each term in the first binomial by each term in the second binomial.
• In the given exercise, we tackled the multiplication using the difference of squares formula. This is a specialized method where the binomials fit the pattern \((a-b)(a+b)=a^2-b^2\).
• The advantage here is that it simplifies calculations significantly by only requiring you to find the squares of the first and second term respectively, then subtract the second from the first.

Overall, mastering binomial products allows for deeper understanding and efficiency in handling more complex algebraic expressions.

Algebraic Expressions

Algebraic expressions like \((x - 3y)\) and \((x + 3y)\) are composed of variables and constants. They often include operators like addition, subtraction, multiplication, and division. When working with algebraic expressions, there are several key elements to consider:

• Variables represent unknown quantities, such as x and y in the expression.
• Constants refer to fixed numbers, like 3 in our example.
• Each term in the expression is separated by a plus or minus sign.

The beauty of algebraic expressions lies in their ability to represent real-world problems in a mathematical form. By manipulating these expressions, you can solve equations, factor expressions, and find values of unknown variables. In our problem, the combination of the terms in sequential operations helped factorize the expression efficiently.

Factoring Techniques

Factoring techniques allow you to decompose algebraic expressions into simpler, multipliable forms. This skill is crucial in solving equations and simplifying expressions. There are several methods in factoring, each useful in different scenarios.

• The difference of squares technique is particularly helpful when you have expressions like \((x-3y)(x+3y)\).By identifying patterns, this technique simplifies calculations through direct application of the formula \(a^2 - b^2 = (a-b)(a+b)\), which saves time and effort.
• It's vital to recognize common patterns in expressions where this formula can be applied. Make sure the expression fits \((a-b)(a+b)\) which immediately allows for simplification to the difference of squares.
• This technique not only simplifies expressions but also helps in solving equations where expressing a statement in simpler forms is key.

Mastering factoring equips you with tools to tackle a variety of problems, making complex equations more manageable and approachable.