Problem 116 - Introductory Algebra for College Students

Question

Perform the indicated operation. $$(9 x+10)(9 x-10)$$

Step-by-Step Solution

Answer

The simplified form of the expression \( (9x + 10)(9x - 10) \) is \( 81x^2 - 100 \).

Step 1: Recognize the pattern

The given expression \( (9x + 10)(9x - 10) \) can be seen as a difference of two squares. That's because it has the form of \( (a + b)(a - b) \) which is equal to \( a^2 - b^2 \). In this case, \( a \) is \( 9x \) and \( b \) is \( 10 \).

Step 2: Apply the difference of squares

Using the formula \( (a + b)(a - b) = a^2 - b^2 \), to apply it on the expression, it will be \( (9x)^2 - (10)^2 \).

Step 3: Simplify the results

Upon simplifying, \( (9x)^2 \) equals \( 81x^2 \) and \( (10)^2 \) equals \( 100 \). So, the simplified form of the expression is \( 81x^2 - 100 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are crucial elements in mathematics as they allow us to represent numbers and operations symbolically. These expressions consist of variables (like \(x\)), constants (like numbers), and operational symbols (such as \(+\), \(-\), \(\times\)). In the exercise provided, the algebraic expression
\[(9x + 10)(9x - 10)\]
is structured to help us explore advanced techniques by composing simple operations into a complex expression.A powerful understanding of algebraic expressions enables students to manipulate and simplify even complicated equations. This skill is foundational in solving equations, performing algebraic operations, and preparing for polynomial manipulations.

• Variables: Stand-ins for unknown quantities, which in this case is \(x\).
• Constants: Numbers that don't change, like 9 and 10 in the expression.
• Operations: Include addition, subtraction, multiplication, which combine these numbers and variables into expressions.

Polynomial Operations

Polynomial operations involve handling polynomials, which are algebraic expressions with one or more terms. In general, these operations include addition, subtraction, multiplication, and division of polynomials.
In our exercise, you are tasked with multiplying two binomials. A binomial is a polynomial with exactly two terms.When multiplying polynomials, the distributive property is often used, but in the case of the expression \((9x + 10)(9x - 10)\), we use a shortcut due to its special form. This shortcut is known as the difference of squares.

• Difference of Squares: Special product pattern where \((a + b)(a - b) = a^2 - b^2\).
• Efficiency: Recognizing patterns like the difference of squares simplifies multiplication, saving time and effort in computations.

Recognizing such forms not only speeds up the process but helps develop an intuition for algebraic structures.

Factoring Techniques

Factoring is an essential skill in algebra, especially when simplifying expressions or solving equations. The goal of factoring is to write an expression as a product of simpler expressions, making it easier to work with.
In the example provided, applying a pattern recognition technique simplifies the process using the difference of squares.Understanding the concept of factoring techniques in algebraic expressions is fundamental. Many factoring techniques exist, ranging from factoring out a common factor to recognizing quadratic and cubic factorizations. However, the difference of squares is a specific and frequently encountered technique:

• Recognizing Patterns: Identify when an expression can be factored using known patterns like \((a + b)(a - b)\).
• Application: Use the formula \(a^2 - b^2\) to simplify expressions, as shown in the solution where \((9x)^2 - (10)^2\) becomes \(81x^2 - 100\).
• Practicality: Such techniques are not just theoretical. They are applied frequently in fields requiring solution of polynomial equations like physics and engineering.

By mastering these techniques, students enhance their problem-solving toolkit, making algebraic manipulations more intuitive and efficient.