Which Expression Is A Factor Of Both X^2-9 And X^2+8x+15?

Jul 16, 2025 by ADMIN 58 views

Introduction

In this article, we will delve into the question: "Which expression is a factor of both x^2 - 9 and x^2 + 8x + 15?" This is a common type of problem encountered in algebra, particularly in the context of factoring polynomials. Mastering such problems is crucial for success in higher mathematics. We will explore the underlying concepts, step-by-step solutions, and provide additional insights to ensure a thorough understanding. Factoring polynomials is a fundamental skill in algebra. It involves breaking down a polynomial expression into simpler expressions (factors) that, when multiplied together, yield the original polynomial. This process is essential for solving equations, simplifying expressions, and understanding the behavior of functions. The ability to identify common factors between different polynomials is particularly useful in various mathematical contexts, including simplifying rational expressions and solving systems of equations. This article aims to provide a detailed explanation of how to solve this problem, reinforcing key algebraic principles and problem-solving techniques.

Understanding the Problem

To address the question effectively, we need to understand the core concepts of factoring quadratic expressions. A quadratic expression is a polynomial of degree two, generally in the form ax^2 + bx + c, where a, b, and c are constants. Factoring a quadratic expression involves finding two binomials (expressions with two terms) that, when multiplied together, result in the original quadratic expression. In this case, we have two quadratic expressions: x^2 - 9 and x^2 + 8x + 15. Our task is to find a common factor that exists in both expressions. This requires us to factor each expression individually and then identify the factors that are present in both. Understanding the structure of quadratic expressions and the techniques for factoring them is essential for solving this type of problem. Factoring is not just a mechanical process; it involves recognizing patterns and applying algebraic principles. By mastering these techniques, you can simplify complex expressions and solve a wide range of mathematical problems. This article will guide you through the factoring process for each expression, highlighting the key steps and strategies involved. We will also discuss common pitfalls to avoid and provide tips for efficient problem-solving.

Factoring x^2 - 9

The first expression, x^2 - 9, is a classic example of the difference of squares. The difference of squares is a special factoring pattern that takes the form a^2 - b^2, which can be factored into (a + b)(a - b). Recognizing this pattern is key to factoring this expression efficiently. In our case, x^2 is the square of x, and 9 is the square of 3. Therefore, we can rewrite the expression as x^2 - 3^2. Applying the difference of squares formula, we get: x^2 - 9 = (x + 3)(x - 3). This factorization tells us that the factors of x^2 - 9 are (x + 3) and (x - 3). Understanding the difference of squares pattern is crucial for factoring certain quadratic expressions quickly and accurately. This pattern is widely used in algebra and calculus, making it an essential concept to master. By recognizing this pattern, you can avoid more complex factoring methods and arrive at the solution more efficiently. In this case, applying the difference of squares pattern directly leads to the factorization of x^2 - 9, making the subsequent steps of identifying common factors much simpler.

Factoring x^2 + 8x + 15

The second expression, x^2 + 8x + 15, is a trinomial. Factoring a trinomial of the form x^2 + bx + c involves finding two numbers that add up to b and multiply to c. In this case, we need to find two numbers that add up to 8 and multiply to 15. Let's consider the pairs of factors of 15: 1 and 15, 3 and 5. The pair 3 and 5 satisfies both conditions: 3 + 5 = 8 and 3 * 5 = 15. Therefore, we can factor the trinomial as follows: x^2 + 8x + 15 = (x + 3)(x + 5). This factorization tells us that the factors of x^2 + 8x + 15 are (x + 3) and (x + 5). Factoring trinomials is a fundamental skill in algebra, and it often requires some trial and error. However, by systematically considering the factors of the constant term and their sums, you can efficiently find the correct factorization. This technique is widely applicable in solving quadratic equations and simplifying algebraic expressions. The ability to factor trinomials accurately is essential for success in higher-level mathematics. In this case, identifying the correct pair of factors (3 and 5) allows us to easily factor the expression and proceed to find the common factor.

Identifying the Common Factor

Now that we have factored both expressions, we can identify the common factor. We found that:

x^2 - 9 = (x + 3)(x - 3)
x^2 + 8x + 15 = (x + 3)(x + 5)

Comparing the factors of both expressions, we can see that the common factor is (x + 3). This means that (x + 3) is a factor of both x^2 - 9 and x^2 + 8x + 15. Identifying common factors is a crucial step in simplifying expressions and solving equations. It allows us to reduce complex expressions to simpler forms, making them easier to work with. In this case, recognizing the common factor of (x + 3) not only answers the question but also demonstrates the power of factoring in identifying shared components between different expressions. This skill is particularly useful in simplifying rational expressions and solving systems of equations. The ability to quickly and accurately identify common factors is a valuable asset in any algebraic problem-solving scenario.

Analyzing the Answer Choices

Now, let's analyze the answer choices provided in the original question:

A. (x + 5) B. (x + 3) C. (x - 3) D. (x - 9)

Based on our analysis, we have determined that the common factor of both expressions is (x + 3). Therefore, the correct answer choice is B. (x + 3). Analyzing the answer choices in the context of the factored expressions is a crucial step in verifying the solution. It ensures that the identified common factor aligns with the available options. This step reinforces the understanding of the problem and the accuracy of the solution. In this case, by comparing the factors we found with the answer choices, we can confidently select the correct option. This process highlights the importance of careful analysis and attention to detail in mathematical problem-solving.

Conclusion

In conclusion, the expression that is a factor of both x^2 - 9 and x^2 + 8x + 15 is (x + 3). This was determined by factoring both expressions individually and identifying the common factor. This problem highlights the importance of understanding factoring techniques, particularly the difference of squares and factoring trinomials. Mastering these techniques is essential for success in algebra and higher mathematics. The ability to factor polynomials efficiently and accurately is a valuable skill that will benefit you in various mathematical contexts. By understanding the underlying concepts and practicing these techniques, you can confidently tackle similar problems and achieve a deeper understanding of algebra. This article has provided a comprehensive guide to solving this problem, reinforcing key algebraic principles and problem-solving strategies. Remember, practice and consistent effort are key to mastering these concepts.

Which Expression is a Factor of Both x^2-9 and x^2+8x+15? Solution and Explanation