Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In Exercises, factor the polynomial. If the polynomial is prime, state it. $$ 4 a^{2} b^{2}-25 c^{2} $$

Step-by-Step Solution

Answer

The factored form of the polynomial \(4a^2b^2 - 25c^2\) is \((2ab - 5c)(2ab + 5c)\).

Step 1: Identify the difference of squares pattern

In the given polynomial \(4a^2b^2 - 25c^2\), we notice that both terms are perfect squares: - The first term is \(4a^2b^2 = (2ab)^2\) - The second term is \(-25c^2 = (-5c)^2\) The polynomial can be written as \((2ab)^2 - (-5c)^2\), which is a difference of squares pattern.

Step 2: Apply the difference of squares identity

Since we've identified the polynomial as a difference of squares, we can apply the difference of squares identity: \(a^2 - b^2 = (a - b)(a + b)\) In this case, we have \(a = 2ab\) and \(b = 5c\), so the polynomial can be factored as: \((2ab - 5c)(2ab + 5c)\)

Step 3: Write the final factored form

Now that we've successfully factored the polynomial, we can write it in its factored form: \(4a^2b^2 - 25c^2 = (2ab - 5c)(2ab + 5c)\) The polynomial has been factored.

Key Concepts

Difference of Squares

In algebra, the "difference of squares" is a specific pattern where two perfect squares are subtracted from each other. This pattern is quite common, and recognizing it can greatly simplify many algebraic expressions. For example, in our exercise, the polynomial \(4a^2b^2 - 25c^2\) exhibits this pattern.

Each term in a difference of squares is a square itself. Let's break that down:

• The term \(4a^2b^2\) is a perfect square because it can be rewritten as \((2ab)^2\).
• Similarly, \(25c^2\) is a perfect square because it's equivalent to \((5c)^2\).

The formula used for factoring a difference of squares is \(a^2 - b^2 = (a - b)(a + b)\). Applying this to our polynomial gives us \((2ab - 5c)(2ab + 5c)\). Recognizing and applying this pattern makes factoring easier and more efficient, especially with polynomials involving high powers or multiple variables.

Factoring Techniques

Factoring is the process of breaking down an expression into a product of simpler expressions, called factors. Different techniques exist for factoring, and recognizing the appropriate one can streamline your problem-solving process.

One key technique is identifying common factors. For polynomials like \(4a^2b^2 - 25c^2\), look for patterns such as the difference of squares where each term is a perfect square. This approach uses the formula \(a^2 - b^2 = (a - b)(a + b)\), simplifying the expression quickly without extensive calculations.

• Understand the properties of exponents and square roots, since they help identify perfect squares.
• Practice re-expressing terms to fit known patterns, such as converting \(4a^2b^2\) to \((2ab)^2\).

Being adept at different factoring methods will improve your ability to solve complex algebraic expressions swiftly and accurately.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators—such as addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions is fundamental in algebra, and it provides a solid basis for further mathematical concepts.

Consider the expression \(4a^2b^2 - 25c^2\) from our exercise. This expression involves variables \(a, b,\) and \(c\), and uses exponents to express the powers of these variables. Recognizing expressions that can be factored, such as the difference of squares, empowers you to simplify or solve equations effectively.

• Understand each component: This helps in identifying potential patterns or factoring opportunities.
• Use algebraic identities: These simplify the task of factoring complex expressions, like the difference of squares which reduces the original problem to a much simpler form.

Building a robust understanding of algebraic expressions enables one to approach problems systematically and uncover solutions with greater confidence and clarity.