Problem 25 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Formula

The given problem involves multiplying two binomials: \((x+3)(x-3)\). This follows the special product formula for the difference of squares: \((a+b)(a-b) = a^2 - b^2\).

Step 2: Identify Components

Identify the values of \(a\) and \(b\) in the expression \((x+3)(x-3)\). Here, \(a = x\) and \(b = 3\).

Step 3: Apply the Formula

Apply the difference of squares formula by plugging \(a = x\) and \(b = 3\) into the formula \(a^2 - b^2\). This gives \(x^2 - 3^2\).

Step 4: Perform the Calculations

Calculate the squares of the terms: \(x^2\) remains as it is, and \(3^2 = 9\). Therefore, the expression simplifies to \(x^2 - 9\).

Step 5: Write the Final Product

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

Key Concepts

Binomial Multiplication

Binomial multiplication is an essential concept in algebra, especially when dealing with polynomial expressions. A binomial is an algebraic expression containing two terms connected by either addition or subtraction. Multiplification of binomials involves using the distributive property. When you come across an expression like \((x+3)(x-3)\), you are multiplying two separate binomials to create a single polynomial. The primary method to handle such multiplication is the distributive property, also known as expanding the brackets:

• Multiply each term in the first binomial by each term in the second binomial.
• Combine like terms if necessary.

Yet, when faced with specific forms like \((x+3)(x-3)\), it is advantageous to apply the 'difference of squares' formula, which provides a quicker path to the simplified form.

Special Product Formula

The special product formula for the difference of squares is an efficient way to multiply binomials of the form \((a+b)(a-b)\). This particular formula allows us to bypass the usual steps associated with the distributive property. Instead, it directly gives the result: \((a+b)(a-b) = a^2 - b^2\).Understanding how this works is key:

• The terms \(a+b\) and \(a-b\) are conjugates, meaning they share a common structure but have opposite operations (+ and -).
• Using this formula simplifies the entire process into finding and subtracting the square of the second term from the square of the first.

For example, with \((x+3)(x-3)\), just identify \(a = x\) and \(b = 3\) to apply the formula and get \(x^2 - 9\) directly.

Polynomial Simplification

Polynomial simplification involves taking a polynomial expression and rewriting it in a simpler or more compact form. After using the special product formula to find the product of the binomials \((x+3)(x-3)\), you end up with \(x^2 - 9\), which is already a simple polynomial expression.
To simplify polynomials you've determined:

• Express the polynomial in terms of like terms.
• Reduce it by performing operations that lead to a simpler form if needed.

In this specific case of the difference of squares, the calculation \(x^2 - 9\) is already in its simplest form, but knowing how to identify and apply simplification strategies is crucial for handling more complex polynomial expressions in the future.