Problem 51 - Elementary and Intermediate Algebra

Question

Find each special product. $$ (9 x+6)(9 x-6) $$

Step-by-Step Solution

Answer

The special product is \(81x^2 - 36\).

Step 1: Recognize the Formula

Notice that the expression \((9x+6)(9x-6)\) fits the pattern of the difference of squares formula, which is \((a+b)(a-b) = a^2 - b^2\). Here, \(a = 9x\) and \(b = 6\).

Step 2: Substitute and Simplify

Substitute \(a = 9x\) and \(b = 6\) into the formula and simplify. This results in: \((9x)^2 - (6)^2 = 81x^2 - 36\).

Step 3: Write the Final Expression

Combine the simplified terms to get the final expression. Therefore, the special product is \(81x^2 - 36\).

Key Concepts

Difference of Squares

The difference of squares is a special algebraic pattern that emerges when multiplying two binomials of the form

• \((a + b)(a - b)\)

This results in the expression, \[a^2 - b^2\]. This formula arises because the middle terms cancel each other out when expanded.
For instance, consider the original problem \((9x + 6)(9x - 6)\). Here, the terms inside the parentheses conveniently fit the difference of squares structure, where

• \(a = 9x\) and
• \(b = 6\).

Applying the difference of squares formula, we replace

• \(a^2\) with \((9x)^2 = 81x^2\),
• and \(b^2\) with \(6^2 = 36\),

resulting in \(81x^2 - 36\). Understanding this pattern simplifies solving similar algebraic expressions and extends beyond to numerous equations.

Special Products

Special products in algebra refer to the result of multiplying specific pairs of polynomial expressions. These include the

• difference of squares,
• perfect square trinomials, and
• cubes of a binomial.

Identifying these special products is crucial for efficiently solving algebraic problems.
The difference of squares is just one example of a special product, which simplifies directly into a subtraction of squared terms \(a^2 - b^2\). This approach can save time and avoid errors since the process avoids the need for lengthy multiplication processes.
In the given exercise, using the special product formula helps us reach the answer faster, transforming

• \((9x+6)(9x-6)\) into \(81x^2 - 36\).

With repeated practice, recognizing these patterns becomes intuitive, enabling quicker calculations and deeper understanding of polynomial behavior.

Polynomial Expressions

Polynomials are mathematical expressions that consist of terms, which are comprised of constants and variables raised to whole number powers. Such an expression can be simple, like

• \(3x + 2\),
• or more complex like \(8x^3 + 5x^2 + 7x + 1\).

A polynomial's degree is determined by the highest power of the variable.
The exercise involved simplifying a product of two binomials that is a polynomial itself. This simplification was made possible by recognizing it as a special product, which turned the task into a difference of squares expression.
Understanding polynomials is fundamental in algebra, as they form the basis for equations, inequalities, and many real-world applications. Emphasizing the understanding of polynomial structure allows for the identification and manipulation of special products, ultimately simplifying the solution process.