Graph Behavior Of Polynomial Function F(x) = X⁵ - 6x⁴ + 9x³

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Polynomial functions are fundamental in mathematics, and understanding their graphical behavior is crucial for solving various problems. In this article, we will delve into how to analyze the graph of a polynomial function, focusing on its roots and their multiplicities. We will use the specific example of the polynomial function f(x) = x⁵ - 6x⁴ + 9x³ to illustrate the key concepts. By examining its factored form, we can determine the points where the graph intersects or touches the x-axis and how the graph behaves at these points. This analysis will provide a comprehensive understanding of how the graph of this polynomial function behaves.

Analyzing the Polynomial Function f(x) = x⁵ - 6x⁴ + 9x³

To understand the behavior of the polynomial function f(x) = x⁵ - 6x⁴ + 9x³, the first step is to factorize it. Factoring the polynomial allows us to identify its roots, which are the values of x for which f(x) = 0. These roots are critical points that determine where the graph intersects or touches the x-axis. By finding these roots and their multiplicities, we can accurately describe the graph's behavior. Let's begin by factoring the given polynomial.

Factoring the Polynomial

The given polynomial function is f(x) = x⁵ - 6x⁴ + 9x³. To factor this polynomial, we first look for common factors in all terms. In this case, x³ is a common factor. Factoring out x³ gives us:

f(x) = x³(x² - 6x + 9)

Now, we need to factor the quadratic expression x² - 6x + 9. This quadratic expression is a perfect square trinomial, which can be factored as follows:

x² - 6x + 9 = (x - 3)²

Thus, the completely factored form of the polynomial function is:

f(x) = x³(x - 3)²

Identifying the Roots and Their Multiplicities

From the factored form f(x) = x³(x - 3)², we can identify the roots and their multiplicities. The roots are the values of x that make the polynomial equal to zero. These roots are x = 0 and x = 3.

  1. Root x = 0: The factor x³ indicates that x = 0 is a root with multiplicity 3. The multiplicity of a root affects how the graph behaves at that point. An odd multiplicity means the graph will cross the x-axis, while an even multiplicity means the graph will touch the x-axis and turn around.
  2. Root x = 3: The factor (x - 3)² indicates that x = 3 is a root with multiplicity 2. Since the multiplicity is even, the graph will touch the x-axis at x = 3 but not cross it.

Understanding the multiplicities of the roots is crucial for sketching the graph and determining its behavior at key points. In the next section, we will discuss how these multiplicities influence the graph's shape.

Impact of Multiplicity on Graph Behavior

The multiplicity of a root significantly influences the behavior of the graph of a polynomial function at that root. As we identified earlier, the polynomial function f(x) = x³(x - 3)² has a root at x = 0 with multiplicity 3 and a root at x = 3 with multiplicity 2. Let's explore how these multiplicities affect the graph's behavior.

  1. Root x = 0 with Multiplicity 3: Since the multiplicity is odd (3), the graph crosses the x-axis at x = 0. Furthermore, because the multiplicity is greater than 1, the graph not only crosses the x-axis but also has a point of inflection at this root. This means the graph flattens out near x = 0 before crossing the axis. The higher the odd multiplicity, the flatter the graph becomes near the x-axis before crossing.
  2. Root x = 3 with Multiplicity 2: Because the multiplicity is even (2), the graph touches the x-axis at x = 3 but does not cross it. Instead, the graph turns around at this point. This behavior is typical of roots with even multiplicities, where the graph approaches the x-axis, touches it, and then moves away in the same direction it came from. This point is also a local extremum, either a local maximum or a local minimum.

By understanding the impact of multiplicity, we can sketch a more accurate graph of the polynomial function. In the next section, we will synthesize our analysis to describe the overall behavior of the graph.

Describing the Graph's Behavior

Based on our analysis, we can now describe the behavior of the graph of the polynomial function f(x) = x⁵ - 6x⁴ + 9x³. We have determined the roots and their multiplicities, which are critical for understanding how the graph interacts with the x-axis. Let's summarize our findings:

  1. Roots: The polynomial has roots at x = 0 and x = 3.
  2. Multiplicity at x = 0: The root x = 0 has a multiplicity of 3, which is odd. This means the graph crosses the x-axis at x = 0, and there is a point of inflection, causing the graph to flatten out near this point.
  3. Multiplicity at x = 3: The root x = 3 has a multiplicity of 2, which is even. This means the graph touches the x-axis at x = 3 but does not cross it. The graph turns around at this point, indicating a local extremum.

Overall Graph Behavior

Considering the roots and their multiplicities, we can infer the following about the graph's behavior:

  • At x = 0, the graph crosses the x-axis and has a point of inflection.
  • At x = 3, the graph touches the x-axis and turns around.

Therefore, the correct statement describing the graph of the polynomial function f(x) = x⁵ - 6x⁴ + 9x³ is:

The graph crosses the x-axis at x = 0 and touches the x-axis at x = 3.

This comprehensive analysis demonstrates how factoring a polynomial, identifying its roots, and understanding their multiplicities can provide a clear picture of the graph's behavior. By following these steps, we can accurately describe and sketch the graphs of various polynomial functions.

Conclusion

In this article, we have explored how to analyze the graph of a polynomial function by factoring it, identifying its roots, and understanding the impact of their multiplicities. Using the example of f(x) = x⁵ - 6x⁴ + 9x³, we demonstrated how the factored form, f(x) = x³(x - 3)², reveals the roots x = 0 and x = 3, with multiplicities 3 and 2, respectively. The multiplicity of a root determines whether the graph crosses or touches the x-axis, and this information is crucial for sketching the graph accurately.

The graph of f(x) = x⁵ - 6x⁴ + 9x³ crosses the x-axis at x = 0 because the multiplicity is odd, and it touches the x-axis at x = 3 because the multiplicity is even. This detailed analysis underscores the importance of understanding the relationship between the algebraic representation of a polynomial function and its graphical behavior. By applying these principles, one can analyze and describe the graphs of a wide range of polynomial functions, enhancing their understanding of polynomial behavior in mathematics.

This approach not only aids in solving mathematical problems but also provides a deeper insight into the nature of functions and their graphical representations. Polynomial functions are foundational in various fields, including engineering, physics, and computer science, making this knowledge highly valuable for anyone pursuing these disciplines. Understanding their behavior allows for accurate modeling and prediction in many real-world applications.