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The difference of squares is a special type of algebraic expression, structured as the subtraction of two square terms. This can be understood as a formula: \(a^2 - b^2 = (a - b)(a + b)\). It's essential because it provides a quick way to factor expressions of this kind. Here, \(a\) and \(b\) are placeholders for any expression that is being squared. In our given exercise, since \((15-r)^2 - s^2\) fits the structure of a difference of squares, we can identify \(a\) as \(15 - r\) and \(b\) as \(s\).

• Identify terms: Determine what is being squared—here, \((15-r)\) and \(s\).
• Use the identity: Apply \((a - b)(a + b)\) to factor the expression efficiently.

Recognizing this pattern greatly simplifies the factoring process and is crucial for solving polynomial equations efficiently.

Algebraic expressions are combinations of variables, numbers, and operators (like addition or subtraction) that represent real-world quantities or scenarios. They form the basis of algebra and can be simple or complex, ranging from single-variable equations to multi-variable polynomials.In our exercise, the expression \((15-r)^2 - s^2\) involves both subtraction and exponentiation, making it a little more complex. Here are some key points to understand about algebraic expressions:

• Variables and Constants: In our case, \(r\) and \(s\) are variables that can take any value, while 15 is a constant number.
• Operations: This includes basic arithmetic and power operations (squared terms here).
• Structure: Knowing how the terms are arranged helps in identifying patterns like the difference of squares.

Understanding and manipulating these expressions lets you solve equations, model scenarios and even predict outcomes in various applications.

Walking through the step-by-step solution of the given exercise ensures clarity and a deeper understanding of each stage in the factoring process.1. **Recognize the Pattern:** - Identify that the expression resembles a difference of squares, a common pattern that simplifies factoring. - Recall \((15-r)^2 - s^2\) is similar to \(a^2 - b^2\) with \(a = 15 - r\) and \(b = s\).2. **Apply the Formula:** - Plug \(a\) and \(b\) into the difference of squares formula: \((a - b)(a + b)\). - For our terms: \(((15 - r) - s)((15 - r) + s)\).3. **Simplify the Result:** - Perform the arithmetic operation within each factor to arrive at the correct result: \((15 - r - s)\) and \((15 - r + s)\).4. **Write the Factored Form:** - Combine these results to get the fully factored expression: \((15 - r - s)(15 - r + s)\).Each step builds on the previous one, ensuring that the final product is both accurate and easy to comprehend. These steps are universally applicable to a wide range of algebraic expressions that fit certain patterns, making the approach both practical and versatile.