Finding Zeros And Multiplicities Of F(x)=-x^5+9x^4-18x^3

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Hey guys! Let's dive into the fascinating world of polynomial functions and uncover the secrets hidden within their graphs. Today, we're going to tackle a specific problem that involves finding the zeros of a polynomial. Zeros, also known as roots or x-intercepts, are the points where the graph of a function intersects the x-axis. Finding these zeros is crucial for understanding the behavior and characteristics of the function.

Our mission is to determine the zeros of the polynomial function f(x) = -x^5 + 9x^4 - 18x^3. We'll explore how to factor this polynomial, identify its zeros, and determine their multiplicities. So, buckle up and get ready to unravel the mystery of this function's zeros!

Factoring the Polynomial

To find the zeros of the polynomial function, the first crucial step involves factoring. Factoring breaks down the polynomial into simpler expressions, making it easier to identify the values of x that make the function equal to zero. Let's dive into the process of factoring our polynomial, f(x) = -x^5 + 9x^4 - 18x^3.

Our initial focus should be on identifying common factors among the terms. In this case, we observe that each term contains a power of x. Specifically, the lowest power of x present in all terms is x^3. Additionally, we can factor out a -1 to simplify the expression further. This leads us to:

f(x) = -x3(x2 - 9x + 18)

Now, we have successfully factored out -x^3, leaving us with a quadratic expression inside the parentheses: x^2 - 9x + 18. To complete the factoring process, we need to factor this quadratic expression. We're on the lookout for two numbers that, when multiplied, give us 18 and, when added, give us -9. After some thought, we can identify these numbers as -3 and -6.

Therefore, we can factor the quadratic expression as (x - 3)(x - 6). Putting it all together, the fully factored form of our polynomial function is:

f(x) = -x^3(x - 3)(x - 6)

This factored form is a treasure map! It reveals the zeros of the function and their multiplicities. By setting each factor equal to zero, we can pinpoint the x-values where the function crosses the x-axis. Let's move on to the next section and extract these valuable zeros from our factored polynomial.

Identifying the Zeros

Now that we've successfully factored the polynomial function into f(x) = -x^3(x - 3)(x - 6), we're in a prime position to identify its zeros. Remember, zeros are the x-values that make the function equal to zero. In our factored form, each factor corresponds to a potential zero.

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

  • -x^3 = 0 => x = 0
  • x - 3 = 0 => x = 3
  • x - 6 = 0 => x = 6

So, we've discovered the zeros of our polynomial function! They are 0, 3, and 6. These are the x-values where the graph of the function intersects the x-axis. But there's more to the story than just the zeros themselves. We also need to understand their multiplicities.

Multiplicity refers to the number of times a particular factor appears in the factored form of the polynomial. This tells us about the behavior of the graph near that zero. Does it simply cross the x-axis, or does it bounce off it? To answer these questions, we need to examine the exponents of each factor in our factored polynomial. Let's move on to the next section to unravel the concept of multiplicity and how it affects the graph of our function.

Determining Multiplicities

We've identified the zeros of our polynomial function as 0, 3, and 6. Now, let's delve into the concept of multiplicity, which adds another layer of understanding to the behavior of the function's graph near these zeros. Multiplicity tells us how many times a particular factor appears in the factored form of the polynomial.

Looking back at our factored polynomial, f(x) = -x^3(x - 3)(x - 6), we can determine the multiplicity of each zero by examining the exponent of its corresponding factor:

  • The factor -x^3 corresponds to the zero x = 0. The exponent of this factor is 3, so the zero 0 has a multiplicity of 3.
  • The factor (x - 3) corresponds to the zero x = 3. The exponent of this factor is 1 (since it's not explicitly written, we assume it's 1), so the zero 3 has a multiplicity of 1.
  • The factor (x - 6) corresponds to the zero x = 6. The exponent of this factor is also 1, so the zero 6 has a multiplicity of 1.

Now, what does this multiplicity tell us about the graph of the function? Here's the key:

  • Odd Multiplicity: If a zero has an odd multiplicity (like 1 or 3), the graph of the function crosses the x-axis at that zero. Think of it as the graph passing straight through the x-axis.
  • Even Multiplicity: If a zero has an even multiplicity (like 2 or 4), the graph of the function touches the x-axis but doesn't cross it. It bounces off the x-axis at that zero, changing direction without going to the other side.

In our case, the zero 0 has a multiplicity of 3 (odd), so the graph crosses the x-axis at x = 0. The zeros 3 and 6 both have a multiplicity of 1 (odd), so the graph also crosses the x-axis at x = 3 and x = 6.

Conclusion

Alright, guys, we've successfully navigated the world of polynomial zeros and multiplicities! We started with the polynomial function f(x) = -x^5 + 9x^4 - 18x^3 and embarked on a journey to uncover its hidden zeros.

We learned the crucial skill of factoring polynomials, breaking down complex expressions into simpler components. By factoring out common factors and factoring quadratic expressions, we transformed our polynomial into its factored form: f(x) = -x^3(x - 3)(x - 6).

From this factored form, we identified the zeros of the function: 0, 3, and 6. These are the points where the graph of the function intersects the x-axis. But we didn't stop there! We delved deeper into the concept of multiplicity, understanding how it affects the behavior of the graph near these zeros.

We discovered that the zero 0 has a multiplicity of 3, while the zeros 3 and 6 each have a multiplicity of 1. This means the graph crosses the x-axis at all three zeros. The multiplicity of 3 at x=0 indicates a more complex crossing behavior, where the graph might flatten out slightly before crossing.

So, to recap, the zeros of the graph of f(x) = -x^5 + 9x^4 - 18x^3 are:

  • 0 with multiplicity 3
  • 3 with multiplicity 1
  • 6 with multiplicity 1

By understanding how to factor polynomials, identify zeros, and determine their multiplicities, we gain a powerful toolkit for analyzing and understanding the behavior of polynomial functions and their graphs. Keep exploring, and you'll uncover even more fascinating insights into the world of mathematics!

I hope this article has clarified how to determine the zeros of a polynomial function and their multiplicities. Keep practicing, and you'll become a pro at finding those zeros!