Problem 37 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(3 x-4)(3 x+4)$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Special Product Formula

The expression \((3x-4)(3x+4)\) fits the form of a difference of squares, which uses the formula \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 3x\) and \(b = 4\).

Step 2: Apply the Formula

Using the difference of squares formula, we substitute \(a = 3x\) and \(b = 4\) into \(a^2 - b^2\). This gives us \((3x)^2 - (4)^2\).

Step 3: Simplify the Expression

Compute \((3x)^2\) to get \(9x^2\) and \((4)^2\) to get \(16\). This simplifies the expression to \(9x^2 - 16\).

Step 4: Conclusion

The simplified form of the original expression \((3x-4)(3x+4)\) using the special product formula is \(9x^2 - 16\).

Key Concepts

Special Product Formulas

Special product formulas are a set of formulas that help us quickly and accurately multiply certain types of algebraic expressions. These formulas save time and reduce the potential for errors during multiplication. One of the most important special product formulas is the "difference of squares" formula.

The difference of squares formula is \[(a-b)(a+b) = a^2 - b^2\] This formula is very useful because it allows us to multiply expressions without actually expanding two binomials into four terms.

• The term \(a\) represents one part of the expression,
• while \(b\) represents the other part of the expression.

In the example expression \((3x-4)(3x+4)\), we can see it's composed in this special way.
The special product formula provides a quicker route, as it only requires squaring \(a\) and \(b\) and then subtracting, rather than using the distributive property to expand and combine terms.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and arithmetic operations that are structured together. These expressions can range from simple to complex and are foundational in algebraic operations.

An algebraic expression can look like \(3x - 4\) or \(5y^2 + 3y - 8\).
An important aspect of understanding algebraic expressions is recognizing that they are often manipulated using various algebraic techniques and formulas. This manipulation often involves operations such as:

• Substitution, where values are replaced in place of variables,
• Simplification, which involves reducing expressions to their simplest form, and
• Factorization, where expressions are rewritten as a product of their factors.

In the exercise, the expressions \(3x - 4\) and \(3x + 4\) individually represent binomials, which means they are made up of two terms. Recognizing patterns within expressions can lead to the efficient application of relevant formulas or methods, such as the difference of squares.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing the expression to its simplest form. This process makes expressions easier to work with and understand.

For example, when simplifying the expression \((3x-4)(3x+4)\) using the difference of squares formula, we break it down to \(9x^2 - 16\).
Simplifying involves a combination of strategies, including:

• Combining like terms, which are terms with the same variable and power.
• Using identities or formulas like the difference of squares, which help in reducing the number of terms.
• Reducing fractions, if any are present in the expression.

By applying these strategies and tools like the special product formulas, we make "simplified" expressions that are easier to understand and further work with, enhancing our overall problem-solving abilities in algebra.