Finding Root Multiplicity: A Step-by-Step Guide

Hey guys! Let's dive into a cool math concept: figuring out the multiplicity of roots in a function. In simple terms, the multiplicity of a root tells us how many times a particular value makes the function equal to zero. We'll break down the function $k(x) = x(x+2)^3(x+4)^2(x-5)^4$, a pretty complex one. Don't worry; it's easier than it looks. This guide will walk you through it step by step. Understanding root multiplicity is super useful in calculus for sketching graphs and understanding function behavior. So, let's get started!

What is Root Multiplicity?

Okay, first things first: what exactly do we mean by the multiplicity of a root? When we talk about a 'root' of a function, we're referring to the values of x that make the function, k(x) in our case, equal to zero. The multiplicity of a root indicates how many times that root appears in the factored form of the function.

Think of it like this: if a root has a multiplicity of 1, it means the corresponding factor appears once. If it has a multiplicity of 2, it appears twice, and so on. The higher the multiplicity, the more the function 'touches' or 'bounces off' the x-axis at that point. For instance, a root with an even multiplicity (like 2 or 4) will cause the graph to touch the x-axis and 'bounce' back. An odd multiplicity (like 1 or 3) will cause the graph to cross the x-axis at that point. Knowing the multiplicity gives you a good idea of what the graph of the function looks like. It tells you a lot about its behavior around these key points, so it's super important.

Breaking Down the Function

Now, let's look at our function: $k(x) = x(x+2)^3(x+4)^2(x-5)^4$. This is already beautifully factored for us, which makes our job way easier! Each factor gives us a root. Remember, the roots are the values of x that make each factor equal to zero. We need to find each root and then figure out its multiplicity, which comes directly from the exponent of the factor. It's just that simple! So, let's work on it.

We'll work through each part of the function and explain how to find the root and its multiplicity. This is fundamental stuff that applies to so many different kinds of math problems, so pay attention, and you'll do great! Also, keep in mind that the function is already in a factored form, which is like a secret weapon in your arsenal. We don't need to factor it ourselves. This is the most important part, understanding that the exponent tells us the multiplicity.

Identifying the Roots and Their Multiplicities

Alright, time to get down to business and actually figure out those multiplicities! Let's go through the function $k(x) = x(x+2)^3(x+4)^2(x-5)^4$ term by term, shall we?

Analyzing the first factor: x

First, we have the factor x. This is the same as (x - 0). To find the root, we set this factor equal to zero: x = 0. Because x has an implied exponent of 1 (it's like x¹), the multiplicity of the root 0 is 1. So the graph of the function will cross the x-axis at x=0.

Analyzing the second factor: (x+2)³

Next, we've got (x+2)³. To find the root, we set x + 2 = 0, which gives us x = -2. The exponent on this factor is 3. Therefore, the multiplicity of the root -2 is 3. This means the graph will cross the x-axis at x = -2. The function doesn't just pass through; it will 'flatten out' a bit as it crosses, thanks to that cube!

Analyzing the third factor: (x+4)²

Now, let's look at (x+4)². Setting x + 4 = 0, we find the root x = -4. The exponent on this factor is 2. Thus, the multiplicity of the root -4 is 2. Since the multiplicity is even, the graph will 'bounce' off the x-axis at x = -4.

Analyzing the fourth factor: (x-5)⁴

Lastly, we've got (x-5)⁴. Setting x - 5 = 0, we get the root x = 5. The exponent on this factor is 4. Consequently, the multiplicity of the root 5 is 4. The graph will touch the x-axis at x = 5 and 'bounce' off, but it'll do so more 'flatly' than at a root with a multiplicity of 2, thanks to the fourth power.

Summary of the Results

So, here's a neat summary of what we've found:

• Root 0: Multiplicity of 1
• Root -2: Multiplicity of 3
• Root -4: Multiplicity of 2
• Root 5: Multiplicity of 4

Knowing these multiplicities gives us key insights into the graph's behavior. For example, we know the graph crosses the x-axis at x=0 and x=-2, and bounces at x=-4 and x=5. This is incredibly helpful for sketching the function. You're getting really close to mastering this concept; just keep up the good work.

Understanding the Graph's Behavior

Alright, now that we've calculated the multiplicities, let's talk a bit about what this means for the graph of the function. Remember, the multiplicity tells us how the graph behaves around each root.

• Multiplicity of 1 (like at x=0): The graph will simply cross the x-axis at this point. It's like a straight line going through the axis. Nothing fancy.
• Multiplicity of 2 (like at x=-4): The graph will touch the x-axis and then 'bounce' back. It doesn't cross; it just gently turns around and goes back in the direction it came.
• Multiplicity of 3 (like at x=-2): The graph crosses the x-axis, but it does so with a bit of a 'flattening' effect. It almost looks like it hesitates before going through.
• Multiplicity of 4 (like at x=5): The graph touches the x-axis and bounces back, but it does so in a way that's even flatter than a multiplicity of 2. It's a very smooth bounce.

This information helps us sketch the graph accurately. For example, if we know where the roots are and their multiplicities, we can quickly draw the shape of the function without plotting many points. It's a huge time-saver and gives you a great understanding of how the function behaves.

Why Multiplicity Matters in the Real World

You might be thinking,