Polynomial Function With Leading Coefficient 2, Root -4 And Root 10

by Pedro Alvarez 68 views

Hey guys! Ever find yourself staring at a polynomial function question and feeling totally lost? Don't worry, we've all been there. Today, we're going to break down a common type of problem: figuring out which polynomial function matches specific clues about its roots and leading coefficient. Let's dive into a question that'll help us understand this better: Which polynomial function has a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1?

Understanding the Clues: Leading Coefficient, Roots, and Multiplicity

Before we jump into solving the problem, let's make sure we're all on the same page about what these clues actually mean. The clues given in this question are the leading coefficient, the roots, and their multiplicities. Understanding these will help us to construct the correct polynomial function.

The Leading Coefficient

The leading coefficient is the number that's multiplied by the term with the highest power of x in our polynomial. It's a big deal because it tells us about the function's end behavior – what happens to the function as x gets really, really big (positive or negative). In our case, the leading coefficient is 2. This means the term with the highest power of x will have a coefficient of 2. This is a crucial piece of information because it helps narrow down our options when looking at potential polynomial functions. A leading coefficient of 2 indicates that the polynomial will likely open upwards (if the highest power of x is even) or have a generally positive trend as x increases (if the highest power of x is odd). So, keep an eye out for this number in the equations – it's like the function's signature!

Roots: Where the Function Touches Zero

Next up, we have roots. Sometimes, you'll hear them called zeros or x-intercepts. Basically, roots are the values of x that make the polynomial function equal to zero. Graphically, these are the points where the function's graph crosses or touches the x-axis. We're told our polynomial has a root of -4 and a root of 10. This tells us that if we were to graph this polynomial, it would definitely intersect the x-axis at x = -4 and x = 10. Remembering that each root corresponds to a factor in the polynomial is key. If -4 is a root, then (x + 4) must be a factor, and if 10 is a root, then (x - 10) is a factor. This connection between roots and factors is a cornerstone of building polynomial functions.

Multiplicity: How Many Times a Root Appears

Now, let's talk about multiplicity. This is where things get a little more interesting. The multiplicity of a root tells us how many times that root appears as a factor in the polynomial. It affects how the graph of the function behaves at that x-intercept. Our problem states that -4 has a multiplicity of 3. This means the factor (x + 4) appears three times: (x + 4)(x + 4)(x + 4), or (x + 4)³. The root 10 has a multiplicity of 1, so the factor (x - 10) appears just once. Multiplicity is super important because it changes the way the graph interacts with the x-axis. A root with a multiplicity of 1 will typically cross the x-axis, while a root with a multiplicity of 2 will touch the x-axis and bounce back. A root with a multiplicity of 3, like ours, will create a sort of flattened S-shape at the x-axis. Understanding multiplicity gives us a deeper insight into the polynomial's graph and behavior.

Building the Polynomial: Putting the Clues Together

Okay, now that we've got a solid grip on what leading coefficients, roots, and multiplicities mean, let's get to the fun part: building our polynomial function! We're going to use the information we have to piece together the correct equation. This process is like solving a puzzle, where each clue fits into a specific place to reveal the final picture. The main goal here is to translate the information about roots and their multiplicities into factors, and then combine those factors with the leading coefficient to form the polynomial function. Remember, each root corresponds to a factor, and the multiplicity tells us how many times that factor appears. The leading coefficient ensures that the polynomial's end behavior matches what is expected.

From Roots to Factors

First, let's focus on the roots. We know that a root of -4 corresponds to a factor of (x + 4). Since -4 has a multiplicity of 3, this factor will appear three times: (x + 4)(x + 4)(x + 4), or (x + 4)³. This means that the graph of the function will have a special behavior at x = -4, sort of flattening out as it crosses the x-axis due to the odd multiplicity. For the root 10, which has a multiplicity of 1, the corresponding factor is (x - 10). This factor appears only once, indicating that the graph will simply cross the x-axis at x = 10 without any flattening or bouncing. Each factor plays a crucial role in shaping the polynomial function, and understanding how roots translate into factors is a key step in building the equation.

Incorporating the Leading Coefficient

Now, don't forget about our leading coefficient! We were told it's 2, which means the entire polynomial will be multiplied by 2. This ensures that the function has the correct vertical stretch. Think of the leading coefficient as the function's overall scaling factor. It doesn't change the roots, but it does affect how steeply the function increases or decreases. Including the leading coefficient is essential for getting the polynomial function exactly right. Without it, the shape of the graph might be correct, but the vertical scaling would be off.

The Complete Polynomial Function

So, let's put it all together. We have our leading coefficient of 2, the factor (x + 4)³ from the root -4 with multiplicity 3, and the factor (x - 10) from the root 10 with multiplicity 1. Combining these gives us the polynomial function: f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10) or, more compactly, f(x) = 2(x + 4)³(x - 10). This is the equation that satisfies all the given conditions. It has the correct leading coefficient, the specified roots, and the correct multiplicities. This final form represents the complete polynomial function that we were aiming to find. It incorporates all the clues provided and gives us a clear picture of the function's behavior.

Analyzing the Options: Which One Matches?

Alright, we've built our polynomial function from scratch. Now, let's compare it to the options given in the problem to see which one matches. This is a critical step because it confirms that our understanding of leading coefficients, roots, and multiplicities is accurate. By carefully examining each option, we can reinforce our learning and ensure we've correctly applied the concepts. This step is also a great way to double-check our work and catch any potential errors before settling on a final answer.

Comparing Our Function to the Choices

We're looking for an equation that looks like f(x) = 2(x + 4)³(x - 10). Let's break down the options:

  • Option A: f(x) = 2(x - 4)(x - 4)(x - 4)(x + 10). This option has a leading coefficient of 2, which is correct, but the factors are (x - 4) and (x + 10). This would give us roots of 4 and -10, which are not what we're looking for. The signs are flipped, indicating that this option does not match our criteria.
  • Option B: f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10). This one also has a leading coefficient of 2, which is a good start. The factors are (x + 4)³ and (x - 10). This gives us roots of -4 (with multiplicity 3) and 10 (with multiplicity 1), exactly what we need! This option looks promising as it aligns perfectly with our derived polynomial function.
  • Option C: f(x) = 3(x - 4)(x - 4)(x + 10). This option has a leading coefficient of 3, which doesn't match our given leading coefficient of 2. Also, the factors (x - 4) and (x + 10) would give us roots of 4 and -10, respectively, which are incorrect. The leading coefficient and the roots both disqualify this option.

The Verdict

After carefully comparing each option to our constructed polynomial function, it's clear that Option B is the correct answer. It has the leading coefficient of 2, a root of -4 with multiplicity 3 [(x + 4)³], and a root of 10 with multiplicity 1 [(x - 10)]. This option perfectly matches all the clues provided in the problem.

Final Answer: Option B is the Winner!

So, there you have it! The polynomial function with a leading coefficient of 2, a root of -4 with multiplicity 3, and a root of 10 with multiplicity 1 is f(x) = 2(x + 4)(x + 4)(x + 4)(x - 10), which is Option B. We nailed it by breaking down the problem into smaller, manageable steps and carefully considering each clue. Remember, understanding leading coefficients, roots, and multiplicities is key to solving these types of problems. Keep practicing, and you'll become a polynomial pro in no time!

Polynomial functions can seem tricky at first, but by understanding the core components—leading coefficients, roots, and their multiplicities—we can confidently construct and identify the correct equation. Keep practicing, and soon you'll be solving these problems with ease!