Finding Multiplicity: Root X=2 In Polynomial Y = X³ - 3x² + 4

Hey guys! Today, we're diving into the exciting world of polynomials, specifically focusing on how to determine the multiplicity of a root. We'll be tackling the polynomial y = x³ - 3x² + 4 and figuring out the multiplicity of the root x = 2. Trust me, it's not as scary as it sounds! This is a fundamental concept in algebra and calculus, crucial for understanding the behavior of polynomial functions and their graphs. So, let’s break it down step by step, making sure everyone gets a solid grasp on this topic. Whether you're a student prepping for an exam, a math enthusiast eager to learn more, or just someone curious about the inner workings of polynomials, this guide is for you. Let's get started and unravel this mathematical mystery together!

Understanding Multiplicity

Before we jump into the problem, let's quickly recap what multiplicity actually means. In simple terms, the multiplicity of a root tells us how many times a particular root appears as a solution of the polynomial equation. For example, if a root has a multiplicity of 2, it means the factor corresponding to that root appears twice in the factored form of the polynomial. This affects how the graph of the polynomial behaves at that root; a root with an even multiplicity (2, 4, etc.) will cause the graph to touch the x-axis and bounce back, while a root with an odd multiplicity (1, 3, etc.) will cause the graph to cross the x-axis. Understanding multiplicity is super important for graphing polynomials and solving related problems. Remember, guys, a root can have a multiplicity of 1 (meaning it appears once), 2 (appearing twice), 3 (appearing thrice), and so on. Let’s visualize this with a few quick examples. If we have a polynomial that factors to (x - 2)(x - 3), both roots, 2 and 3, have a multiplicity of 1. But if the polynomial factors to (x - 2)²(x - 3), the root 2 has a multiplicity of 2, and the root 3 still has a multiplicity of 1. This difference significantly impacts the shape of the graph near x = 2. So, with the concept of multiplicity fresh in our minds, let's apply this knowledge to our specific problem.

Verifying that x=2 is a Root

Okay, so the first thing we need to do is confirm that x = 2 is indeed a root of the polynomial y = x³ - 3x² + 4. How do we do that? Easy peasy! We just substitute x = 2 into the polynomial and see if it equals zero. If it does, then x = 2 is definitely a root. Let's plug it in: y = (2)³ - 3(2)² + 4. Now, let's simplify: y = 8 - 3(4) + 4. Further simplification gives us: y = 8 - 12 + 4. And finally, y = 0. Awesome! So, we've confirmed that x = 2 is a root of the polynomial. But remember, just knowing it's a root isn't enough; we need to find its multiplicity. This is where things get a little more interesting. We've taken the first step by verifying that when x equals 2, the polynomial y evaluates to zero. This confirms that x = 2 is a solution, or a root, of the equation. However, the multiplicity tells us more about how this root behaves in the context of the entire polynomial. Does the graph simply pass through the x-axis at x = 2, or does it touch the x-axis and turn around? The multiplicity will give us that crucial information. So, now that we know x = 2 is a root, let's proceed to find out just how many times this root appears in the polynomial, which will reveal its multiplicity.

Finding the Multiplicity Using Synthetic Division

Alright, guys, now for the fun part – finding the multiplicity. One of the most efficient ways to do this is by using synthetic division. If you're not familiar with synthetic division, it's a neat little shortcut for dividing a polynomial by a linear factor (like x - 2). Since we know x = 2 is a root, this means (x - 2) is a factor of our polynomial. We're going to use synthetic division to divide our polynomial by (x - 2) and see what we get. Here’s how it works: First, write down the coefficients of our polynomial y = x³ - 3x² + 4. Notice that we have a missing x term (there's no x¹ term), so we need to include a 0 as its coefficient. Our coefficients are therefore 1, -3, 0, and 4. Now, set up the synthetic division table. Write the root (2) outside the table and the coefficients inside. Bring down the first coefficient (1). Multiply it by the root (2) and write the result under the next coefficient (-3). Add those two numbers. Multiply the result by the root (2) again and write it under the next coefficient (0). Add those numbers. Repeat this process one more time. The last number you get is the remainder. If the remainder is 0 (which it should be since x = 2 is a root), then the other numbers are the coefficients of the quotient polynomial. If you perform the synthetic division once and get a remainder of 0, it means (x - 2) is a factor. To find the multiplicity, we need to check if (x - 2) is a factor again. So, we’ll perform synthetic division on the quotient polynomial we just obtained. If we get a remainder of 0 again, then (x - 2) appears twice as a factor, meaning the root x = 2 has a multiplicity of at least 2. We keep repeating this process until we get a non-zero remainder, which tells us how many times (x - 2) is a factor. This method is super useful because it not only helps us find the multiplicity but also breaks down the polynomial into smaller, more manageable factors.

Performing Synthetic Division: A Step-by-Step Example

Okay, let's walk through the synthetic division process step by step with our polynomial y = x³ - 3x² + 4 and the root x = 2. First, we write down the coefficients: 1 (for x³), -3 (for -3x²), 0 (for the missing x term), and 4 (the constant term). Now, set up your synthetic division table. You'll have the root (2) on the left and the coefficients 1, -3, 0, and 4 inside the table. 1. Bring Down the First Coefficient: Bring down the first coefficient, which is 1, below the line. 2. Multiply and Add: Multiply the number you just brought down (1) by the root (2), which gives you 2. Write this 2 under the next coefficient (-3). Add -3 and 2, which gives you -1. Write -1 below the line. 3. Repeat: Multiply -1 by the root (2), which gives you -2. Write -2 under the next coefficient (0). Add 0 and -2, which gives you -2. Write -2 below the line. 4. Final Step: Multiply -2 by the root (2), which gives you -4. Write -4 under the last coefficient (4). Add 4 and -4, which gives you 0. Write 0 below the line. That last number, 0, is our remainder. Since the remainder is 0, we know that (x - 2) is indeed a factor. The other numbers we got (1, -1, -2) are the coefficients of the quotient polynomial. This means after the first division, we have x² - x - 2. Now, here’s the crucial part: to find the multiplicity, we need to see if (x - 2) is a factor of this new polynomial as well. So, we perform synthetic division again using the quotient polynomial's coefficients (1, -1, -2) and the root 2. Let’s run through this second round of synthetic division, which will help us nail down the multiplicity once and for all.

Determining the Multiplicity

Let's continue with our second round of synthetic division. This time, we're using the coefficients of the quotient polynomial we got from the first division: 1 (for x²), -1 (for -x), and -2 (the constant term). Again, we'll use the root x = 2. Set up the synthetic division table with 2 on the left and the coefficients 1, -1, and -2 inside. 1. Bring Down the First Coefficient: Bring down the first coefficient, which is 1, below the line. 2. Multiply and Add: Multiply the number you just brought down (1) by the root (2), which gives you 2. Write this 2 under the next coefficient (-1). Add -1 and 2, which gives you 1. Write 1 below the line. 3. Final Step: Multiply 1 by the root (2), which gives you 2. Write 2 under the last coefficient (-2). Add -2 and 2, which gives you 0. Write 0 below the line. Once again, our remainder is 0! This is fantastic news because it means (x - 2) is a factor of the quotient polynomial x² - x - 2 as well. So, (x - 2) appears as a factor at least twice. The new quotient we get from this second division has coefficients 1 and 1, which corresponds to the polynomial x + 1. Now, we have our polynomial factored as (x - 2)²(x + 1). We can clearly see that the factor (x - 2) appears twice, which means the root x = 2 has a multiplicity of 2. Guys, we've done it! We've successfully found the multiplicity of the root x = 2. But just to be absolutely sure, let's think about why we don't need to do synthetic division a third time. If we tried to divide (x + 1) by (x - 2), we would definitely get a non-zero remainder. This confirms that (x - 2) is not a factor of (x + 1), and therefore, the multiplicity of x = 2 is exactly 2.

Final Answer: The Multiplicity of x=2

So, after walking through the steps of verifying the root and using synthetic division, we've arrived at our final answer. The multiplicity of the root x = 2 for the polynomial y = x³ - 3x² + 4 is 2. This means that the factor (x - 2) appears twice in the factored form of the polynomial, which we found to be (x - 2)²(x + 1). Remember, a root with a multiplicity of 2 means the graph of the polynomial will touch the x-axis at x = 2 and bounce back, rather than crossing through it. This is a key piece of information when graphing polynomials and understanding their behavior. We can confidently say that we've not only found the multiplicity but also understand its significance in the context of the polynomial function. Hopefully, this step-by-step guide has made the process clear and easy to follow. Next time you encounter a similar problem, you'll be well-equipped to tackle it. Keep practicing, guys, and you'll become masters of polynomial multiplicity in no time!