Problem 82 - Algebra A Combined Function

Question

Multiply. See Section 5.6. \((2 x-9)(2 x+9)\)

Step-by-Step Solution

Answer

The product is \(4x^2 - 81\).

Step 1: Identify the Formula

We recognize the expression \((2x - 9)(2x + 9)\) as being in the form of a difference of squares. The difference of squares formula is \((a - b)(a + b) = a^2 - b^2\). Here, \(a = 2x\) and \(b = 9\).

Step 2: Apply the Formula

Using the formula \((a - b)(a + b) = a^2 - b^2\), we substitute \(a = 2x\) and \(b = 9\) to calculate \((2x)^2 - 9^2\).

Step 3: Calculate Squared Terms

Calculate \((2x)^2\) which is \(4x^2\), and \(9^2\) which is \(81\). This utilizes the power property that \((kx)^2 = k^2 x^2\).

Step 4: Subtract the Squares

Subtract the square of \(b\) from the square of \(a\). This results in \(4x^2 - 81\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together. To multiply polynomials, you need to use the distributive property, commonly known as the FOIL method for binomials. In general, you distribute each term in the first polynomial to every term in the second polynomial.
For example, when dealing with two binomials, such as \((a + b)(c + d)\), you multiply:

• First: Multiply the first terms: \(a \times c\)
• Outer: Multiply the outer terms: \(a \times d\)
• Inner: Multiply the inner terms: \(b \times c\)
• Last: Multiply the last terms: \(b \times d\)

After which, you'll combine all these terms. The result is a new polynomial that usually has more terms unless it is simplified.
In the case of our exercise, \((2x - 9)(2x + 9)\), recognizing and using algebraic identities can simplify the multiplication process.

Algebraic Identities

Algebraic identities are specific, well-known equations that hold true for all variable values. They are useful shortcuts when multiplying polynomials because they simplify calculations and reduce the chance of error. One of the most important identities is the difference of squares. This identity states that for any two numbers, a and b, \((a - b)(a + b) = a^2 - b^2\).
In the given problem, the binomials \((2x - 9)(2x + 9)\) fit this identity perfectly. Here, you can let \(a = 2x\) and \(b = 9\). Recognizing this allows us to directly apply the identity: calculate \((2x)^2\) and \(9^2\), and then subtract to find the result.
The power of using algebraic identities is in the rapid simplification they provide, transforming potentially time-consuming multiplication into much shorter and manageable calculations.

Factoring

Factoring is a technique used to break down complex algebraic expressions into simpler components. It allows us to express a polynomial as the product of other polynomials, usually making it easier to solve or simplify further.
In the context of our original problem, we can understand the reverse process of what was applied. Recognizing that \(4x^2 - 81\) is a difference of squares helps you factor it back into its binomial components: \((2x - 9)(2x + 9)\).

• Identify the 'a' and 'b' in the expression \(a^2 - b^2\).
• Express it as a product: \((a - b)(a + b)\).

This is particularly useful when tasked with solving equations or simplifying expressions, as factoring often reveals solutions or simplifies substitution.