Finding Zeros And Multiplicity: A Step-by-Step Guide

by SLV Team 53 views

Hey guys! Let's dive into finding the zeros and their multiplicities for the function f(x) = x⁶ - 4x⁴. This is a classic problem in algebra, and understanding it is key to grasping polynomial behavior. We'll break down the process step-by-step, making it super easy to follow. Ready? Let's go!

Step 1: Factoring the Polynomial

Our primary goal in this problem is to find the zeros, or the values of x that make f(x) = 0. The first step is always to factor the polynomial. Factoring helps us break down the equation into simpler parts, making it easier to identify the zeros. Let's get started with our function: f(x) = x⁶ - 4x⁴. We can see that both terms have x to some power, so we can factor out the greatest common factor (GCF), which in this case is x⁴. Doing this, we get: f(x) = x⁴(x² - 4). Now, we have a product of two terms: x⁴ and (x² - 4). But wait, we're not done yet! The term (x² - 4) is a difference of squares, which we can further factor into (x - 2)(x + 2). So, our fully factored polynomial becomes f(x) = x⁴(x - 2)(x + 2). Nice, right? We've successfully factored our polynomial! Now, let's use this to find our zeros.

Detailed Explanation of Factoring

Let's take a closer look at why factoring is so important. By factoring, we're essentially rewriting the polynomial in a way that highlights its structure. When we have a product of factors equal to zero, we know that at least one of those factors must be equal to zero. This is the zero-product property, and it's the key to finding the zeros. Consider the initial expression, x⁶ - 4x⁴. It's not immediately obvious what values of x make this equation true. However, by factoring it into x⁴(x - 2)(x + 2), we've broken it down into smaller, manageable pieces. The term x⁴ is a factor that can be equal to zero if x = 0. The term (x - 2) is a factor that can be equal to zero if x = 2. Finally, the term (x + 2) is a factor that can be equal to zero if x = -2. Factoring allows us to systematically identify these individual solutions. It simplifies the complex equation and makes it much easier to solve. Without factoring, we'd be left trying to guess and check, or using more advanced techniques that are not as straightforward. So, always remember, factoring is your friend!

Step 2: Finding the Zeros

Alright, now that we have our factored polynomial, f(x) = x⁴(x - 2)(x + 2), finding the zeros is a breeze. Remember, the zeros are the values of x that make f(x) = 0. Using the zero-product property (if a product of factors is zero, then at least one of the factors must be zero), we can set each factor equal to zero and solve for x. Here's how it breaks down:

  1. x⁴ = 0: Taking the fourth root of both sides gives us x = 0. This is one of our zeros.
  2. x - 2 = 0: Adding 2 to both sides gives us x = 2. This is another zero.
  3. x + 2 = 0: Subtracting 2 from both sides gives us x = -2. This is our third zero.

So, we've found three zeros: x = 0, x = 2, and x = -2. But we're not quite done yet. We also need to determine the multiplicity of each zero. Let's look at that next!

Zero Product Property in Detail

The zero-product property is a fundamental concept in algebra, essentially stating that if the product of several factors equals zero, then at least one of those factors must be zero. Think of it like this: if you're multiplying a bunch of numbers together and the result is zero, then at least one of the numbers you multiplied had to have been zero. This property is incredibly useful because it allows us to break down complex equations into simpler ones. For example, when we have the factored form of our polynomial, x⁴(x - 2)(x + 2) = 0, we can apply the zero-product property to each factor. This gives us three separate equations: x⁴ = 0, (x - 2) = 0, and (x + 2) = 0. By solving each of these equations, we can find the values of x that satisfy the original equation. In essence, the zero-product property transforms a single, complex equation into several simpler, independent equations, making the problem much easier to solve. It's a crucial tool for finding the zeros of polynomials and understanding their behavior.

Step 3: Determining the Multiplicity of Each Zero

Okay, now for the final piece of the puzzle: finding the multiplicity of each zero. The multiplicity of a zero is the number of times that the corresponding factor appears in the factored form of the polynomial. It tells us how the graph of the function behaves at each zero. In our factored form, f(x) = x⁴(x - 2)(x + 2), we can see the following:

  • The factor x appears four times (because of the x⁴). Therefore, the zero x = 0 has a multiplicity of 4.
  • The factor (x - 2) appears once. Therefore, the zero x = 2 has a multiplicity of 1.
  • The factor (x + 2) appears once. Therefore, the zero x = -2 has a multiplicity of 1.

So, we have:

  • Zero at x = 0 with multiplicity 4
  • Zero at x = 2 with multiplicity 1
  • Zero at x = -2 with multiplicity 1

Analyzing Multiplicity and Graph Behavior

The multiplicity of a zero tells us about the graph's behavior at that point. When a zero has an odd multiplicity (like 1, 3, 5, etc.), the graph crosses the x-axis at that point. This means the function changes signs (from positive to negative or vice versa) as it passes through the zero. For example, the zeros x = 2 and x = -2 both have a multiplicity of 1 (which is odd), so the graph will cross the x-axis at these points. Conversely, when a zero has an even multiplicity (like 2, 4, 6, etc.), the graph touches the x-axis at that point but doesn't cross it. Instead, it bounces off the x-axis. Our function has a zero at x = 0 with a multiplicity of 4 (even). Therefore, the graph of f(x) will touch the x-axis at x = 0 and