Problem 31 - Algebra and Trigonometry

Question

In Exercises 15-58, find each product. $$ (x+3)(x-3) $$

Step-by-Step Solution

Answer

The product of the given binomial expressions \( (x+3)(x-3) \) is \( x^2 - 9 \)

Step 1: Distribute the first terms

First, distribute the term \(x\) from the first binomial to each term in the second binomial to get \(x(x-3)\). This simplifies to \(x^2 - 3x\).

Step 2: Distribute the remaining terms

Next, distribute the term 3 from the first binomial to each term in the second binomial to get \(3(x-3)\). This simplifies to \(3x - 9\)

Step 3: Combine like terms

Add the two results from steps 1 and 2 together. This gives us \((x^2 - 3x) + (3x - 9)\). Combining like terms, -3x and 3x cancel each other. So, we left with \(x^2 - 9\).

Key Concepts

Binomials

A binomial is a type of polynomial that contains exactly two terms. In the example given in the exercise, we have two binomials: \( (x+3) \) and \( (x-3) \). These binomials each consist of a linear term and a constant:

• The first binomial, \((x+3)\), includes the terms "\(x\)" and "3".
• The second binomial, \((x-3)\), includes the terms "\(x\)" and "-3".

Binomials are essential building blocks in algebra. They help us understand basic algebraic operations like addition, subtraction, and especially multiplication. When you multiply binomials, you apply certain properties and techniques, such as the distributive property, to simplify expressions. This simplification helps in solving more complex algebraic equations. It's important to be comfortable with binomials since they appear frequently in math problems.

Distributive Property

The distributive property is a fundamental principle in algebra. It allows us to multiply a term across all terms inside a parenthesis. In the case of binomial multiplication, the distributive property is often applied twice.
In Step 1 of the solution, the term "\(x\)" from the first binomial \((x+3)\) is distributed to each term in the second binomial \((x-3)\). This results in:

• \(x \times x = x^2\)
• \(x \times -3 = -3x\)

This gives us \(x^2 - 3x\).
Step 2 applies the distributive property with the second term from the first binomial, "3", resulting in:

• \(3 \times x = 3x\)
• \(3 \times -3 = -9\)

This results in \(3x - 9\).
Using the distributive property effectively is crucial for simplifying expressions, especially as they become more complex. It is a powerful tool that facilitates the handling of algebraic operations.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variables raised to the same power. In algebra, this step is often necessary after distributing terms. It helps in tidying up expressions and making them much easier to interpret.
In Step 3 of the exercise, you have the expression from the previous steps: \(x^2 - 3x + 3x - 9\). Here, -3x and 3x are like terms since both have the variable "\(x\)" raised to the same power, which is 1. When you combine these like terms, they cancel each other out:

• -3x + 3x = 0

This leaves you with the simplified expression \(x^2 - 9\).
By combining like terms, the expression becomes much clearer and reveals the important features or solutions of the problem. This practice is essential for solving algebraic equations and is a crucial part of most polynomial operations.