Problem 44 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(6-3 x)(6+3 x)$$

Step-by-Step Solution

Answer

The product is \(36 - 9x^2\).

Step 1: Identify the Pattern

The expression \((6 - 3x)(6 + 3x)\) is a binomial multiplication of the form \((a-b)(a+b)\), which is the difference of squares formula. This means it can be simplified to \(a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula \((a-b)(a+b) = a^2 - b^2\) requires identifying \(a\) and \(b\). Here, \(a = 6\) and \(b = 3x\). Apply the formula: \[a^2 - b^2 = (6)^2 - (3x)^2\]

Step 3: Calculate Each Square

Calculate \(6^2\) and \((3x)^2\):- \(6^2 = 36\)- \((3x)^2 = 9x^2\).

Step 4: Substitute and Simplify

Substitute the values back into the formula from Step 2: \[36 - 9x^2\]This is the simplified form of the product of the binomials.

Key Concepts

Binomial Multiplication

Binomial multiplication is a process used in algebra to multiply two binomials together. A binomial is an algebraic expression that contains exactly two terms, which can be numbers, variables, or a combination of both. In the expression provided, \[(6-3x)(6+3x)\], we see two binomials multiplied together. To understand how to multiply these binomials, it's important to recognize common patterns such as the difference of squares.
To multiply binomials in general, one might use the FOIL method, which stands for First, Outer, Inner, and Last terms. However, special patterns like the one in our exercise allow us to simplify the process.

• First, identify the form of the binomial multiplication.
• Use specific algebraic identities to simplify the work.

Recognizing these shortcuts makes binomial multiplication much easier.

Algebraic Identities

Algebraic identities are equations that hold true for every possible value of their variables. They serve as powerful tools in algebraic manipulation, helping to simplify seemingly complex expressions. In the context of our exercise, the identity used is the 'difference of squares.' This is one of the most commonly used algebraic identities.

The difference of squares is expressed as:\[ (a-b)(a+b) = a^2 - b^2 \]This identity tells us that a product of a sum and a difference of the same two terms results in the difference of the squares of those terms. Knowing this identity allows us to bypass more labor-intensive multiplication methods and get straight to simplifying the binomial expression.

• Identify the values of \(a\) and \(b\) in the expression.
• Apply the identity to simplify directly to \(a^2 - b^2\).

This approach makes problem-solving clearer and faster.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to work with and understand. After applying an algebraic identity, we often need to further simplify the resulting expression. In our example, once the difference of squares formula is applied to \[(6-3x)(6+3x) = a^2 - b^2\], we proceed by calculating the squares:

• Compute \(6^2 = 36\).
• Compute \((3x)^2 = 9x^2\).

Substitute these values back into the expression, resulting in \[ 36 - 9x^2 \], which is the simplest form of the product of the binomials. This step is crucial as it provides the most concise and understandable version of the expression, suitable for further use in mathematical calculations or problem-solving scenarios.
Simplifying helps in making larger mathematical problems more manageable.