Finding The Roots Of Polynomial Function F(x) = (x^2 + 6x + 8)(x^2 + 6x + 13)

Hey guys! Today, we're diving into the world of polynomial functions and tackling the question of finding the roots of a specific function. It might sound intimidating, but don't worry, we'll break it down step by step. Our mission is to determine the complete list of roots for the polynomial function f(x) = (x^2 + 6x + 8)(x^2 + 6x + 13). This involves a bit of algebra, but with clear explanations, you'll see it's totally manageable. Let's jump right in and figure out how to solve this! Understanding how to find the roots of polynomial functions is a foundational concept in algebra and calculus, making it crucial for further studies in mathematics and related fields. The roots, also known as zeros, are the values of x that make the function equal to zero. These points are where the graph of the function intersects the x-axis, and they provide valuable information about the function's behavior. For instance, knowing the roots can help us sketch the graph of the function, determine its intervals of increase and decrease, and solve equations involving the function. In real-world applications, polynomial functions are used to model various phenomena, such as the trajectory of a projectile, the growth of a population, and the optimization of business processes. Therefore, mastering the techniques for finding roots is not just an academic exercise but a practical skill that can be applied in diverse fields. So, let's embark on this journey of discovery and unlock the secrets of polynomial functions together!

Understanding the Problem

Before we start crunching numbers, let's make sure we fully grasp what the question is asking. We're given a polynomial function, f(x) = (x^2 + 6x + 8)(x^2 + 6x + 13), and our goal is to find all its roots. Remember, the roots of a function are the values of x that make f(x) equal to zero. Basically, we need to solve the equation (x^2 + 6x + 8)(x^2 + 6x + 13) = 0. To do this, we'll use a cool trick: we'll set each factor equal to zero separately. This works because if the product of two expressions is zero, then at least one of them must be zero. This principle is fundamental to solving polynomial equations and allows us to break down a complex problem into simpler parts. By setting each factor to zero, we create two quadratic equations that we can solve individually. This approach not only simplifies the process but also provides a clear pathway to finding all the roots of the original polynomial function. It's like dissecting a complex puzzle into smaller, more manageable pieces. Once we've solved each quadratic equation, we'll combine the solutions to get the complete set of roots for the given function. This method is widely applicable to polynomials that can be factored, making it a valuable tool in any mathematician's toolkit. So, with a solid understanding of this approach, we're well-equipped to tackle the problem at hand and uncover the hidden roots of our polynomial function.

Solving the First Quadratic Factor

Let's tackle the first factor: x^2 + 6x + 8 = 0. This is a quadratic equation, and we can solve it by factoring. We need to find two numbers that add up to 6 (the coefficient of the x term) and multiply to 8 (the constant term). Those numbers are 2 and 4, right? So, we can rewrite the equation as (x + 2)(x + 4) = 0. Now, set each factor equal to zero: x + 2 = 0 and x + 4 = 0. Solving these gives us x = -2 and x = -4. These are two of the roots of our polynomial function. Factoring is a powerful technique for solving quadratic equations because it breaks down the problem into simpler linear equations. When we factor a quadratic expression, we're essentially reversing the process of expansion, which can often lead to a more intuitive understanding of the roots. However, not all quadratic equations can be easily factored. In such cases, we can turn to other methods like the quadratic formula or completing the square. The quadratic formula, in particular, is a versatile tool that can solve any quadratic equation, regardless of whether it can be factored or not. By mastering these techniques, we gain a comprehensive toolkit for solving quadratic equations and can confidently approach a wide range of problems. So, armed with our factoring skills, we've successfully unearthed two of the roots of our polynomial function. Now, let's move on to the second quadratic factor and see what other secrets it holds.

Solving the Second Quadratic Factor

Now, let's move on to the second factor: x^2 + 6x + 13 = 0. This one looks a bit trickier, doesn't it? We can try factoring, but we won't find any nice whole numbers that add up to 6 and multiply to 13. So, what do we do? We use the quadratic formula! The quadratic formula is our trusty friend for solving any quadratic equation of the form ax^2 + bx + c = 0. It states that the solutions are given by: x = (-b ± √(b^2 - 4ac)) / 2a. In our case, a = 1, b = 6, and c = 13. Plugging these values into the formula, we get: x = (-6 ± √(6^2 - 4 * 1 * 13)) / (2 * 1). Simplifying, we have: x = (-6 ± √(36 - 52)) / 2. Further simplification leads to: x = (-6 ± √(-16)) / 2. Ah, a negative number under the square root! This means we're dealing with complex roots. Remember that √(-1) is defined as i (the imaginary unit). So, √(-16) = 4i. Our equation now becomes: x = (-6 ± 4i) / 2. Dividing both terms in the numerator by 2, we get: x = -3 ± 2i. This gives us two complex roots: x = -3 + 2i and x = -3 - 2i. The quadratic formula is a cornerstone of algebra, providing a reliable method for solving any quadratic equation. It's especially useful when factoring is not straightforward or when the roots are not real numbers. The presence of a negative value under the square root signals that the roots are complex, which opens up a whole new realm of mathematical possibilities. Complex numbers have numerous applications in fields like electrical engineering, quantum mechanics, and signal processing. By mastering the quadratic formula and understanding complex numbers, we expand our problem-solving toolkit and gain the ability to tackle a wider range of mathematical challenges. So, with these complex roots in hand, we've completed our quest to find all the roots of the second quadratic factor. Now, let's gather all our findings and present the complete solution.

Combining the Roots

Alright, we've done the hard work! We found the roots from the first factor to be x = -2 and x = -4. And from the second factor, we got the complex roots x = -3 + 2i and x = -3 - 2i. To get the complete list of roots for the polynomial function f(x) = (x^2 + 6x + 8)(x^2 + 6x + 13), we simply combine these solutions. So, the roots are: -2, -4, -3 + 2i, and -3 - 2i. We did it! Combining the roots from each factor is the final step in solving for the complete set of solutions for the polynomial equation. It's like piecing together the individual solutions to form the big picture. This step reinforces the fundamental principle that the roots of a polynomial function are the values that make the function equal to zero. By identifying all such values, we gain a comprehensive understanding of the function's behavior and its relationship to the x-axis. The combination of real and complex roots highlights the richness and complexity of polynomial functions. While real roots correspond to the points where the graph of the function intersects the x-axis, complex roots do not have a direct graphical representation in the real plane. However, they are crucial for understanding the complete solution set of the equation and have significant implications in various mathematical and scientific contexts. So, with all the roots neatly gathered, we've successfully solved the problem and can confidently move on to the next challenge. This process of finding and combining roots is a testament to the power of algebraic techniques and their ability to unravel complex mathematical problems.

Final Answer

Therefore, the complete list of roots for the polynomial function f(x) = (x^2 + 6x + 8)(x^2 + 6x + 13) is -2, -4, -3 + 2i, -3 - 2i. That matches option C, right? So, C is the correct answer. High five! Finding the correct answer is the culmination of our efforts and a testament to our problem-solving skills. It's a moment of satisfaction and a validation of the techniques we've learned. But beyond just getting the right answer, it's important to appreciate the journey we've taken to get there. We've explored factoring, the quadratic formula, and the concept of complex roots – all essential tools in the mathematician's arsenal. The process of solving this problem has not only enhanced our understanding of polynomial functions but also sharpened our critical thinking and analytical skills. These skills are transferable to a wide range of challenges, both within and outside the realm of mathematics. So, while the final answer is the destination, the journey of problem-solving is equally important. It's where we learn, grow, and develop the abilities that will empower us to tackle future challenges with confidence and competence. Congratulations on reaching the final answer and embracing the learning process along the way!