Multiplicity Of Zeros: Function F(x) Explained

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Multiplicity of Zeros: Function f(x) Explained

Let's dive into understanding the concept of multiplicity of zeros with a specific function. We're going to break down the function f(x) = (x-3)2(x+2)2(x-1) to identify its zeros and their respective multiplicities. Guys, this is a super important concept in algebra and calculus, so let's get right to it!

Understanding Zeros and Multiplicity

First off, what exactly are zeros? In the simplest terms, zeros of a function are the x-values that make the function equal to zero. Graphically, these are the points where the function's graph intersects or touches the x-axis. Finding these zeros is crucial for understanding the behavior of the function.

Now, let's talk about multiplicity. The multiplicity of a zero tells us how many times a particular factor appears in the factored form of the polynomial. It's like a zero having different levels of impact on the function's behavior. Think of it as the zero's power or weight.

  • A zero with a multiplicity of 1 means the graph of the function will pass straight through the x-axis at that point. There's no special behavior, just a clean crossing.
  • A zero with a multiplicity of 2 (or any even number) means the graph will touch the x-axis at that point and bounce back. The graph doesn't cross the x-axis; it just kisses it and turns around. This is often referred to as a turning point or a tangent point.
  • A zero with a multiplicity of 3 (or any odd number greater than 1) means the graph will flatten out as it passes through the x-axis. It's like a gentler crossing compared to a multiplicity of 1, with a slight pause or inflection at the zero.

Understanding these behaviors helps us sketch the graph of a polynomial function without having to plot a million points. It's a neat trick, trust me!

Analyzing the Function f(x) = (x-3)2(x+2)2(x-1)

Okay, let's get our hands dirty with our function: f(x) = (x-3)2(x+2)2(x-1). The beauty of this function is that it's already in factored form, making it super easy to identify the zeros and their multiplicities.

To find the zeros, we simply set each factor equal to zero and solve for x:

  1. (x-3)^2 = 0 => x - 3 = 0 => x = 3
  2. (x+2)^2 = 0 => x + 2 = 0 => x = -2
  3. (x-1) = 0 => x = 1

So, our zeros are x = 3, x = -2, and x = 1. Now, let's figure out their multiplicities:

  • For x = 3, the factor is (x-3)^2. The exponent 2 tells us that the multiplicity of the zero 3 is 2. This means the graph will touch the x-axis at x = 3 and bounce back.
  • For x = -2, the factor is (x+2)^2. Again, the exponent 2 indicates that the multiplicity of the zero -2 is 2. The graph will also touch the x-axis at x = -2 and bounce back.
  • For x = 1, the factor is (x-1). Since there's no exponent written, it's understood to be 1. So, the multiplicity of the zero 1 is 1. The graph will pass straight through the x-axis at x = 1.

See? It's not that scary once you break it down. The factored form is your best friend in these situations!

Answering the Questions

Now that we've done the analysis, let's directly answer the questions:

  • The zero ___ has a multiplicity of 1.

    Based on our analysis, the zero 1 has a multiplicity of 1.

  • The zero -2 has a multiplicity of ___.

    We found that the zero -2 has a multiplicity of 2.

So, there you have it! We've successfully identified the zeros and their multiplicities for the given function. Remember, understanding multiplicity is key to sketching polynomial graphs and analyzing their behavior. Keep practicing, and you'll master it in no time!

Graphical Representation and Behavior

To solidify our understanding, let's think about how these multiplicities affect the graph of f(x). This visual connection is super helpful. We already know the zeros are x = -2, x = 1, and x = 3.

  • At x = -2 (multiplicity 2): The graph touches the x-axis and turns around. Imagine a parabola sitting on the x-axis at x = -2. It's a local extremum – either a maximum or a minimum.
  • At x = 1 (multiplicity 1): The graph passes straight through the x-axis. This is a regular crossing point, nothing fancy happening here.
  • At x = 3 (multiplicity 2): Similar to x = -2, the graph touches the x-axis and turns around. Another local extremum.

Knowing this, we can start to imagine the overall shape of the graph. Since the leading term of the expanded polynomial (if we were to multiply it out) would be x^5 (positive and odd degree), we know the graph will start low on the left, go through our interesting points, and end high on the right. Key takeaway: The degree of the polynomial and the sign of the leading coefficient give us the end behavior of the graph.

The Importance of Factored Form

I can't stress enough how much the factored form helps us. If the function were given in its expanded form (something like x^5 - 5x^4 - 4x^3 + 36x^2 - 27x - 18), it would be significantly harder to find the zeros. Factoring polynomials can be a challenge, but it unlocks so much information about the function's behavior.

If you're faced with a polynomial in expanded form, try your factoring techniques: look for common factors, try grouping, or use the rational root theorem to find potential rational zeros. Once you have it in factored form, you're golden!

Multiplicity and Real-World Applications

Okay, so we've dissected this polynomial, but you might be wondering,