Polynomial Function With Leading Coefficient 2, Root -4 (Multiplicity 3), And Root 10 (Multiplicity 1)
Hey guys! Today, we're diving into the fascinating world of polynomial functions and how to construct them from their roots and leading coefficients. We've got a fun problem to tackle: identifying the polynomial function that has a leading coefficient of 2, a root of -4 with a multiplicity of 3, and a root of 10 with a multiplicity of 1. Sounds like a mouthful, right? But don't worry, we'll break it down step-by-step so you'll be a polynomial pro in no time! Understanding polynomial functions is crucial for various applications in mathematics, engineering, and computer science. These functions describe a wide range of phenomena, from the trajectory of a ball thrown in the air to the growth of a population. By mastering the concepts behind polynomial functions, you'll gain a powerful tool for analyzing and modeling the world around you. So, let's get started and unlock the secrets of these mathematical marvels!
Understanding the Basics of Polynomial Functions
Before we jump into solving the problem, let's quickly review some key concepts about polynomial functions. This will give us a solid foundation to work from and ensure we're all on the same page. First things first, what exactly is a polynomial function? Well, in simple terms, it's a function that can be expressed as a sum of terms, where each term is a constant multiplied by a variable raised to a non-negative integer power. For example, f(x) = 2x^3 - 5x^2 + 3x - 1 is a polynomial function. The highest power of the variable in the polynomial is called the degree of the polynomial, and the constant multiplying the term with the highest power is called the leading coefficient. In our example, the degree is 3 and the leading coefficient is 2. Another important concept is the root of a polynomial function. A root is simply a value of x that makes the function equal to zero. In other words, it's the value of x where the graph of the polynomial intersects the x-axis. For example, if f(x) = x - 2, then the root is x = 2 because f(2) = 0. Finally, we need to understand the concept of multiplicity. The multiplicity of a root tells us how many times that root appears as a factor in the polynomial. If a root has a multiplicity of n, it means that the corresponding factor appears n times in the factored form of the polynomial. For instance, if a polynomial has a root of x = 3 with a multiplicity of 2, then the factor (x - 3) appears twice in the factored form, like this: (x - 3)(x - 3). Got it? Great! Now we're ready to tackle the problem.
Deconstructing the Problem Statement
Okay, let's dissect the problem statement to make sure we fully understand what we're looking for. The question asks us to identify the polynomial function that satisfies three specific conditions. Let's break these down one by one: 1. Leading coefficient of 2: This means that the number multiplying the term with the highest power of x must be 2. This is a crucial piece of information because it narrows down our choices significantly. Remember, the leading coefficient plays a significant role in determining the end behavior of the polynomial function's graph. A positive leading coefficient means the graph will rise to the right, while a negative one means it will fall. The magnitude of the leading coefficient also affects the steepness of the graph. 2. Root -4 with multiplicity 3: This tells us that x = -4 is a root of the polynomial, and it appears three times. This means that the factor (x - (-4)) or (x + 4) will be present three times in the factored form of the polynomial. The multiplicity of a root has a visual representation on the graph of the polynomial. If the multiplicity is odd, the graph will cross the x-axis at that root. If the multiplicity is even, the graph will touch the x-axis but not cross it. 3. Root 10 with multiplicity 1: This tells us that x = 10 is another root of the polynomial, and it appears only once. This means that the factor (x - 10) will be present once in the factored form of the polynomial. A root with a multiplicity of 1 is called a simple root, and the graph of the polynomial will cross the x-axis at that point. Now that we've carefully examined each condition, we have a clear picture of what the polynomial function should look like. It should have a leading coefficient of 2, a factor of (x + 4) raised to the power of 3, and a factor of (x - 10). Let's use this information to construct the polynomial function.
Constructing the Polynomial Function
Now comes the fun part – building our polynomial function! We know that the function must have a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1. Let's translate this information into mathematical expressions. Since -4 is a root with multiplicity 3, we know that (x - (-4)) or (x + 4) appears as a factor three times. This gives us the factor (x + 4)^3. Similarly, since 10 is a root with multiplicity 1, we know that (x - 10) appears as a factor once. So, we have the factor (x - 10). To ensure the leading coefficient is 2, we multiply the entire expression by 2. Putting it all together, we get the polynomial function: f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) or f(x) = 2(x + 4)^3(x - 10). This is the polynomial function that satisfies all the given conditions! It has a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1. We've successfully constructed the polynomial from its roots and leading coefficient. You see, guys, it's like building with Lego bricks – each root and the leading coefficient is a brick, and we're putting them together to form the complete structure, which is the polynomial function. This process of constructing polynomials from their roots is a fundamental skill in algebra and calculus. It allows us to model various real-world phenomena and solve complex problems. By understanding the relationship between roots, multiplicities, and leading coefficients, you can gain a deeper appreciation for the power and elegance of polynomial functions.
Evaluating the Answer Choices
Alright, let's put our detective hats on and compare our constructed polynomial function with the answer choices provided. This is a crucial step to ensure we haven't made any sneaky mistakes along the way. We've determined that the correct polynomial function should be in the form f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) or f(x) = 2(x + 4)^3(x - 10). Now, let's examine each answer choice:
A. f(x) = 2(x - 4)(x - 4)(x - 4)(x + 10): This option has a leading coefficient of 2, which is good. However, it has a root of 4 with multiplicity 3 and a root of -10 with multiplicity 1. This doesn't match our requirements, so it's not the correct answer.
B. f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10): Bingo! This option perfectly matches our constructed polynomial function. It has a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1. This is our winner!
C. f(x) = 3(x - 4)(x - 4)(x + 10): This option has a root of 4 with multiplicity 2 and a root of -10 with multiplicity 1. Also, the leading coefficient is 3, not 2. So, this is definitely not the correct answer. By carefully comparing our constructed polynomial with the answer choices, we've confidently identified the correct option. This process of elimination is a valuable strategy for tackling multiple-choice questions. It allows you to systematically narrow down the possibilities and increase your chances of selecting the right answer. Remember, always double-check your work and make sure your answer aligns with all the given conditions. A little bit of extra scrutiny can save you from making careless errors.
The Correct Answer
After carefully analyzing the problem and evaluating the answer choices, we've arrived at the solution! The polynomial function that has a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1 is:
B. f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) or f(x) = 2(x + 4)³(x - 10)
We nailed it! By understanding the relationship between roots, multiplicities, and leading coefficients, we were able to construct the correct polynomial function and confidently identify the answer. This problem highlights the importance of having a solid grasp of fundamental concepts in algebra. By breaking down complex problems into smaller, manageable steps, you can tackle even the most challenging questions. Remember, practice makes perfect! The more you work with polynomial functions, the more comfortable and confident you'll become. So, keep exploring, keep learning, and keep having fun with math!
Key Takeaways and Further Exploration
Great job, guys! We successfully solved the problem and identified the polynomial function that met all the given criteria. Let's recap some of the key takeaways from this exercise. First, we learned that a polynomial function can be constructed from its roots and leading coefficient. This is a fundamental concept in algebra and has wide-ranging applications. Second, we emphasized the importance of understanding the concept of multiplicity. The multiplicity of a root tells us how many times that root appears as a factor in the polynomial and affects the behavior of the graph at that point. Third, we highlighted the significance of the leading coefficient. The leading coefficient determines the end behavior of the polynomial function and plays a crucial role in shaping its graph. Finally, we demonstrated the power of problem-solving strategies, such as breaking down complex problems into smaller steps and using process of elimination. These strategies can help you tackle even the most challenging questions with confidence. But our journey doesn't end here! There's so much more to explore in the fascinating world of polynomial functions. You can delve deeper into topics such as:
- Graphing polynomial functions: Learn how to sketch the graph of a polynomial function by analyzing its roots, multiplicities, leading coefficient, and end behavior.
- Polynomial division: Master the techniques of long division and synthetic division to divide polynomials and find factors.
- The Remainder Theorem and the Factor Theorem: Discover how these theorems can help you find roots and factors of polynomials.
- Applications of polynomial functions: Explore how polynomial functions are used to model real-world phenomena in various fields, such as physics, engineering, and economics. So, keep your curiosity burning and continue exploring the amazing world of mathematics! There are endless discoveries to be made.