Problem 43 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$(2-5 x)(2+5 x)$$

Step-by-Step Solution

Answer

The product is \(4 - 25x^2\).

Step 1: Recognize Special Product

Identify that the expression \((2 - 5x)(2 + 5x)\) is an example of the difference of squares. The form of a difference of squares is \((a-b)(a+b) = a^2-b^2\).

Step 2: Identify Values for a and b

In the expression \((2 - 5x)(2 + 5x)\), recognize that \(a\) is 2 and \(b\) is \(5x\).

Step 3: Apply the Formula

Apply the difference of squares formula: \(a^2 - b^2 = (2)^2 - (5x)^2\).

Step 4: Calculate Each Square

Calculate \(a^2\) and \(b^2\). Here, \((2)^2 = 4\) and \((5x)^2 = 25x^2\).

Step 5: Subtract the Squares

Subtract the squares to find the product: \(4 - 25x^2\).

Key Concepts

difference of squares

The difference of squares is a powerful and well-known pattern in algebra, used to simplify the multiplication of two specific binomials. It applies to situations where you multiply a binomial by its conjugate, like \((a-b)(a+b)\). Recognizing this pattern allows for quick simplification by following the difference of squares formula: \(a^2 - b^2\).
- **What to look for**: Two terms in the form \((a \pm b)\).- **Why it's useful**: Saves time in calculations without expanding all terms.Familiarity with the difference of squares opens a pathway to simplify expressions promptly and efficiently, especially during tests or when simplifying large algebraic expressions.

binomial multiplication

Binomial multiplication, often introduced through the acronym FOIL, involves expanding the product of two binomials. However, understanding specific patterns, such as the difference of squares, offers a shortcut.Normally, if you multiply two binomials, \((a+b)(c+d)\), you carry out all possible multiplications:
- **F**irst: Multiply the first terms \(a \times c\)- **O**uter: Multiply the outer terms \(a \times d\)- **I**nner: Multiply the inner terms \(b \times c\)- **L**ast: Multiply the last terms \(b \times d\)However, with the difference of squares pattern, you can skip directly to \(a^2 - b^2\), making the multiplication less time-consuming and less prone to calculation errors.

special products

Special products in algebra refer to formulas or patterns that speed up the multiplication process or simplification of expressions. Recognizing these patterns allows you to avoid lengthy calculations and provides insightful shortcuts. Some common types include:

• Square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\)
• Difference of squares: \((a-b)(a+b) = a^2 - b^2\)

These products arise frequently in algebraic manipulations and are important to master for more elaborate problem-solving scenarios. Recognizing a special product pattern drastically simplifies the work you need to do.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. These expressions form the foundation of algebra, allowing representation of relationships and patterns in a concise manner. When working with expressions, it's crucial to understand how to manipulate and simplify them effectively:

• Combine like terms for simplicity.
• Recognize patterns like special products for efficient simplifications.
• Utilize substitution when necessary to evaluate or simplify expressions.

Understanding algebraic expressions and their behavior, such as how they expand during binomial multiplication, gives you the tools to tackle more advanced algebra topics confidently.