Problem 81 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(3 x+4)(3 x-4)$$

Step-by-Step Solution

Answer

The product is \(9x^2 - 16\).

Step 1: Identify the Algebraic Pattern

The expression to be multiplied is \((3x + 4)(3x - 4)\). Notice that this is in the form of \((a + b)(a - b)\), which is a difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula states that \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 3x\) and \(b = 4\).

Step 3: Calculate \(a^2\)

Substitute \(a = 3x\) into \(a^2\), which gives us \((3x)^2 = 9x^2\).

Step 4: Calculate \(b^2\)

Substitute \(b = 4\) into \(b^2\), which gives us \(4^2 = 16\).

Step 5: Subtract \(b^2\) from \(a^2\)

Now that we have \(a^2 = 9x^2\) and \(b^2 = 16\), substitute them back into the formula to get \(9x^2 - 16\).

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves distributing each term in one polynomial to every term in another. Here, when multiplying \((3x + 4)\) by \((3x - 4)\), recognizing this setup as binomials in a special pattern is key to simplifying the task. Rather than multiplying each term separately, identifying patterns like the difference of squares makes the work easier. This specific pattern comes down to multiplying two binomials with the same terms, but opposite signs, \((a + b) (a - b)\). This leads directly to the formula \(a^2 - b^2\), simplifying computations and eliminating extra steps.

Algebraic Patterns

Spotting algebraic patterns is like finding shortcuts in math problems. They save time and reduce errors.
In this exercise, understanding the difference of squares is handy. The pattern; two identical terms with one subtraction and one addition:

• Form: \((a + b)(a - b)\)
• Result: \(a^2 - b^2\)

This framework uses squares of each part, which involves simply squaring the first term and the second term separately, then subtracting the results. Recognizing and applying these patterns in algebra help to simplify complex expressions efficiently.

Simplifying Expressions

The core goal of simplifying expressions is to reduce them to their simplest form without changing their value.
With the expression \((3x+4)(3x-4)\), following the difference of squares rule takes us directly to \(9x^2 - 16\). This step removes intermediate operations like distributing and combining, swiftly leading to the final simplified output. By using shortcuts like difference of squares, one can turn multiple steps into just a couple of calculations, making simplification fast and accurate. Thus, it's essential to not only compute correctly but to also understand which mathematical laws can streamline and refine expressions. Learning to simplify makes solving more extensive algebra problems much more manageable.