Finding The Polynomial Function: A Step-by-Step Guide

Dec 5, 2025 by ADMIN 54 views

Hey guys! Let's dive into the world of polynomial functions and figure out which one fits the description we've got. We're looking for a polynomial that has a leading coefficient of 2, a root at -4 with a multiplicity of 3, and a root at 10 with a multiplicity of 1. It might sound a bit complex at first, but trust me, we'll break it down step by step to make it super clear. This is a common type of problem in algebra, so understanding it will definitely help you ace your math exams and build a solid foundation for more advanced topics. Let's get started!

Understanding the Basics: Polynomials and Their Roots

So, before we jump into the options, let's refresh our memory on what all this stuff means. A polynomial function is basically an expression that involves variables, constants, and exponents, combined using addition, subtraction, and multiplication. Think of it as a mathematical machine that takes an input (x) and spits out an output (f(x)).

The leading coefficient is the number that sits in front of the term with the highest power of 'x'. It's a crucial part because it tells us about the end behavior of the polynomial. If the leading coefficient is positive, the graph of the polynomial will go up on the right side. If it's negative, it'll go down on the right side. In our case, the leading coefficient is 2, which is positive. This helps us narrow down the possibilities.

Now, let's talk about roots. Roots (also known as zeros or x-intercepts) are the values of 'x' where the polynomial function equals zero, i.e., where the graph of the function crosses the x-axis. A root with a multiplicity tells us how many times that root appears. If a root has a multiplicity of 1, the graph crosses the x-axis at that point. If the multiplicity is 2, the graph touches the x-axis and bounces back. And if the multiplicity is 3, the graph flattens out as it crosses the x-axis. This multiplicity information is super important for sketching the graph of the polynomial. In our problem, we have a root at -4 with a multiplicity of 3, and a root at 10 with a multiplicity of 1.

To put it simply, the multiplicity of a root is all about how many times that particular x-value is a solution to the equation f(x) = 0. So, a root with a multiplicity of three, like the one we've got at -4, indicates that (x + 4) is a factor of the polynomial three times. This understanding is key to unlocking the problem. Let's see how we can use this knowledge to solve this problem.

Decoding the Problem: Leading Coefficient, Roots, and Multiplicity

Alright, now that we're all refreshed on the fundamentals, let's apply our knowledge to the problem. We're looking for a polynomial function with the following characteristics:

Leading Coefficient: 2
Root at -4 with multiplicity 3
Root at 10 with multiplicity 1

This means our function will be of the form f(x) = a(x - r1)(x - r2)..., where 'a' is the leading coefficient, and r1, r2, etc. are the roots. Let's build our polynomial piece by piece. We know that the leading coefficient (a) is 2, so we can start by writing f(x) = 2(...).

Next, let's consider the root -4 with a multiplicity of 3. This means that (x - (-4)) or (x + 4) appears as a factor three times. So, we'll have (x + 4)(x + 4)(x + 4) in our function. Finally, we have a root at 10 with a multiplicity of 1. This means that (x - 10) appears as a factor once. Combining everything together, we get: f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10).

Now, let's analyze the options and determine which one matches our findings. Remember, when a root is given, we write the factor as (x - root). If the root is -4, the factor will be (x - (-4)) = (x + 4). The multiplicity of a root determines how many times the factor appears in the polynomial.

Let’s analyze the problem in terms of the factors. The factor (x+4) comes from the root -4. This factor is repeated three times because the root has a multiplicity of 3. And then, there is the factor (x-10), which comes from the root 10. The leading coefficient is simply a number in front of the factor and determines the end behavior. We have all the pieces we need to identify the correct answer.

Analyzing the Answer Choices: Finding the Right Match

Okay, guys, let's take a look at the answer choices. Remember, we are looking for a function of the form f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10). Let's go through the options one by one:

A. f(x) = 2(x - 4)(x - 4)(x - 4)(x + 10) This option has a leading coefficient of 2, which is correct. However, it incorrectly uses roots of 4 and -10. We need roots of -4 with multiplicity 3 and 10 with multiplicity 1. This is not the correct answer.
B. f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) This option has a leading coefficient of 2, which is correct. It correctly uses a root of -4 with a multiplicity of 3, given as (x+4) multiplied three times. It also correctly uses a root of 10 with a multiplicity of 1, given as (x-10). This matches all the requirements. This looks like a winner!
C. f(x) = 3(x - 4)(x - 4)(x + 10) This option has an incorrect leading coefficient of 3, not 2. Also, the root values are wrong; it should be -4 with a multiplicity of 3 and 10 with a multiplicity of 1. This is not the correct answer.
D. f(x) = 3(x + 4)(x + 4)(x - 10) This option also has an incorrect leading coefficient of 3, not 2. This option has a correct root of -4 with a multiplicity of 2, given as (x+4) multiplied twice. This is not the correct answer.

By carefully examining each option and comparing it to what we know about the leading coefficient, roots, and their multiplicities, we can confidently pick the correct answer. The process of elimination also works well here. We can quickly eliminate options that don't match our criteria, making it easier to find the right one.

The Final Answer: Choosing the Correct Polynomial

After a thorough analysis of the options, we can confidently say that Option B: f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) is the correct answer. This function satisfies all the conditions: a leading coefficient of 2, a root at -4 with a multiplicity of 3, and a root at 10 with a multiplicity of 1.

Congratulations! You've successfully solved this problem. You now have a better understanding of how to construct polynomial functions based on their roots, leading coefficients, and multiplicities. This knowledge will be super valuable as you continue to explore the fascinating world of mathematics. Keep practicing, and you'll become a pro at these types of problems in no time. Remember to always break down problems step-by-step and verify that your answer fulfills all the requirements. Good luck!