Analyzing The Polynomial Function F(x) = X^4 - 4x^3 + 4x^2

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This article provides a comprehensive analysis of the polynomial function f(x) = x^4 - 4x^3 + 4x^2. We will explore various aspects of this function, including its end behavior, zeros, graph, and other key characteristics. This detailed examination will provide a strong understanding of polynomial functions and their properties. We will address specific questions related to the function, offering clear explanations and step-by-step solutions. This article aims to be a valuable resource for students, educators, and anyone interested in learning more about polynomial functions.

Understanding Polynomial Functions

Before diving into the specifics of f(x) = x^4 - 4x^3 + 4x^2, let's establish a foundation by understanding the basics of polynomial functions. A polynomial function is a function that can be expressed in the form:

f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0

where:

  • n is a non-negative integer representing the degree of the polynomial.
  • a_n, a_{n-1}, ..., a_1, a_0 are constants called coefficients.
  • a_n is the leading coefficient.

The degree of a polynomial is the highest power of the variable x. The leading coefficient plays a crucial role in determining the end behavior of the polynomial's graph. Polynomial functions are continuous and have smooth curves, making them essential in modeling various real-world phenomena.

Key Concepts in Polynomial Functions

  • Degree: The highest power of the variable in the polynomial. It influences the shape of the graph and the maximum number of turning points.
  • Leading Coefficient: The coefficient of the term with the highest power. It dictates the end behavior of the graph.
  • Zeros (Roots): The values of x for which f(x) = 0. These are the points where the graph intersects the x-axis.
  • Multiplicity: The number of times a particular factor appears in the factored form of the polynomial. It affects the behavior of the graph at the corresponding zero.
  • End Behavior: The behavior of the graph as x approaches positive or negative infinity.

Analyzing f(x) = x^4 - 4x^3 + 4x^2

Now, let's focus on our specific polynomial function, f(x) = x^4 - 4x^3 + 4x^2. This is a quartic polynomial (degree 4) with a leading coefficient of 1. We will analyze its end behavior, find its zeros, and discuss its graph.

a. Determining End Behavior Using the Leading Coefficient Test

The leading coefficient test is a powerful tool for determining the end behavior of a polynomial function. The end behavior describes what happens to the function's values (f(x)) as x approaches positive infinity (x → ∞) and negative infinity (x → -∞). The test relies on two key factors: the degree of the polynomial and the sign of the leading coefficient.

Understanding the Leading Coefficient Test

The leading coefficient test provides the following guidelines:

  1. Even Degree:
    • If the leading coefficient is positive, the graph rises to both the left and the right (as x → -∞, f(x) → ∞ and as x → ∞, f(x) → ∞).
    • If the leading coefficient is negative, the graph falls to both the left and the right (as x → -∞, f(x) → -∞ and as x → ∞, f(x) → -∞).
  2. Odd Degree:
    • If the leading coefficient is positive, the graph falls to the left and rises to the right (as x → -∞, f(x) → -∞ and as x → ∞, f(x) → ∞).
    • If the leading coefficient is negative, the graph rises to the left and falls to the right (as x → -∞, f(x) → ∞ and as x → ∞, f(x) → -∞).

Applying the Test to f(x) = x^4 - 4x^3 + 4x^2

For the polynomial function f(x) = x^4 - 4x^3 + 4x^2:

  • The degree of the polynomial is 4, which is an even number.
  • The leading coefficient is 1, which is positive.

According to the leading coefficient test, since the degree is even and the leading coefficient is positive, the graph of f(x) rises to both the left and the right. This means that as x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) also approaches positive infinity.

In conclusion, based on the leading coefficient test, the graph of f(x) = x^4 - 4x^3 + 4x^2 rises to the left and rises to the right.

Factoring the Polynomial to Find Zeros

To fully understand the behavior of the polynomial function f(x) = x^4 - 4x^3 + 4x^2, finding its zeros is crucial. Zeros, also known as roots or x-intercepts, are the values of x for which f(x) = 0. These points indicate where the graph of the function intersects the x-axis. Determining the zeros involves factoring the polynomial and setting each factor equal to zero.

Factoring Techniques

Factoring polynomials is a fundamental skill in algebra. Several techniques can be employed, including:

  • Greatest Common Factor (GCF): Identifying and factoring out the largest common factor from all terms.
  • Factoring by Grouping: Grouping terms and factoring out common factors within each group.
  • Quadratic Formula: For quadratic polynomials, the quadratic formula provides the roots.
  • Synthetic Division: A method for dividing a polynomial by a linear factor.

Factoring f(x) = x^4 - 4x^3 + 4x^2

Let's factor the polynomial function f(x) = x^4 - 4x^3 + 4x^2:

  1. Identify the GCF: The greatest common factor of the terms x^4, -4x^3, and 4x^2 is x^2. Factoring out x^2, we get:

    f(x) = x2(x2 - 4x + 4)

  2. Factor the Quadratic: The quadratic expression inside the parentheses, x^2 - 4x + 4, is a perfect square trinomial. It can be factored as:

    x^2 - 4x + 4 = (x - 2)(x - 2) = (x - 2)^2

  3. Complete Factored Form: Combining the factored terms, we have:

    f(x) = x^2(x - 2)^2

Determining the Zeros

Now that we have the factored form of f(x) = x^2(x - 2)^2, we can find the zeros by setting each factor equal to zero:

  1. x^2 = 0 implies x = 0 (with multiplicity 2).
  2. (x - 2)^2 = 0 implies x - 2 = 0, which means x = 2 (with multiplicity 2).

Therefore, the zeros of f(x) = x^4 - 4x^3 + 4x^2 are:

  • x = 0 with multiplicity 2
  • x = 2 with multiplicity 2

In summary, factoring the polynomial f(x) = x^4 - 4x^3 + 4x^2 allows us to identify its zeros: x = 0 and x = 2, both with a multiplicity of 2. These zeros are crucial for sketching the graph of the function and understanding its behavior near the x-axis.

Analyzing the Multiplicity of Zeros and Graph Behavior

The multiplicity of a zero plays a significant role in determining the behavior of the graph of a polynomial function at that zero. The multiplicity refers to the number of times a particular factor appears in the factored form of the polynomial. Understanding multiplicity is essential for accurately sketching the graph and interpreting the function's characteristics.

What is Multiplicity?

When a polynomial is factored, each factor corresponds to a zero of the function. The exponent of a factor indicates the multiplicity of the corresponding zero. For instance, in the factored form f(x) = (x - a)^m, the zero x = a has a multiplicity of m.

Impact of Multiplicity on Graph Behavior

The multiplicity of a zero affects how the graph behaves as it approaches and crosses the x-axis:

  1. Odd Multiplicity: If a zero has an odd multiplicity (e.g., 1, 3, 5), the graph crosses the x-axis at that point. The graph passes through the x-axis, changing its sign from positive to negative or vice versa.
  2. Even Multiplicity: If a zero has an even multiplicity (e.g., 2, 4, 6), the graph touches the x-axis at that point but does not cross it. Instead, the graph bounces off the x-axis, remaining on the same side.

Multiplicity in f(x) = x^4 - 4x^3 + 4x^2

Recall the factored form of our polynomial function:

f(x) = x^2(x - 2)^2

From this, we identified the zeros:

  • x = 0 with multiplicity 2
  • x = 2 with multiplicity 2

Both zeros, x = 0 and x = 2, have an even multiplicity of 2. This means that the graph of f(x) will touch the x-axis at these points but will not cross it. The graph will "bounce" off the x-axis at x = 0 and x = 2.

Visualizing the Graph Behavior

At x = 0, the graph touches the x-axis and changes direction without crossing it. This behavior is characteristic of a zero with even multiplicity. Similarly, at x = 2, the graph touches the x-axis and bounces back, staying on the same side of the x-axis. This indicates another zero with even multiplicity.

Understanding the multiplicity of zeros provides valuable insights into the graph's behavior near the x-axis. In the case of f(x) = x^4 - 4x^3 + 4x^2, the even multiplicities at x = 0 and x = 2 tell us that the graph will touch the x-axis at these points but will not cross it, leading to a distinctive "bouncing" effect.

Sketching the Graph of f(x) = x^4 - 4x^3 + 4x^2

Sketching the graph of a polynomial function involves combining information about its end behavior, zeros, and the multiplicity of those zeros. These elements provide a framework for creating an accurate representation of the function's behavior across its domain. For f(x) = x^4 - 4x^3 + 4x^2, we've already established key features that will guide our sketch.

Gathering Key Information

Before sketching, let's summarize the information we've gathered about f(x) = x^4 - 4x^3 + 4x^2:

  1. End Behavior: The graph rises to the left and rises to the right (since the degree is even and the leading coefficient is positive).
  2. Zeros:
    • x = 0 with multiplicity 2
    • x = 2 with multiplicity 2
  3. Multiplicity: Both zeros have even multiplicity, meaning the graph will touch the x-axis at these points but not cross it.

Steps to Sketch the Graph

  1. Plot the Zeros: Mark the zeros on the x-axis. We have zeros at x = 0 and x = 2.

  2. Consider Multiplicity: Since both zeros have a multiplicity of 2, the graph will touch the x-axis at these points and bounce back.

  3. Determine End Behavior: The graph rises to the left and right. This means that as x goes to negative infinity, f(x) goes to positive infinity, and as x goes to positive infinity, f(x) also goes to positive infinity.

  4. Find Additional Points (Optional): To get a more accurate sketch, you can find additional points by plugging in values for x and calculating the corresponding f(x). For example:

    • f(1) = (1)^4 - 4(1)^3 + 4(1)^2 = 1 - 4 + 4 = 1

  5. Sketch the Curve:

    • Start from the left, where the graph rises. Approach the x-axis at x = 0. Since the multiplicity is even, touch the x-axis and bounce back up.
    • Continue until you reach x = 2. Again, touch the x-axis and bounce back up.
    • Finally, continue rising to the right, as the end behavior dictates.

Visual Representation

The graph will resemble a "W" shape, touching the x-axis at x = 0 and x = 2. The curve will be above the x-axis except at these two points.

Key Features of the Sketch

  • The graph touches the x-axis at x = 0 and x = 2. This behavior is characteristic of zeros with even multiplicity.
  • The graph is above the x-axis except at the points where it touches the axis.
  • The graph rises to the left and right, consistent with its end behavior.

By combining the end behavior, zeros, and multiplicity, we can create an accurate sketch of the polynomial function f(x) = x^4 - 4x^3 + 4x^2. This sketch provides a visual representation of the function's behavior and key characteristics.

Conclusion

In this comprehensive analysis, we have thoroughly examined the polynomial function f(x) = x^4 - 4x^3 + 4x^2. We began by applying the leading coefficient test to determine its end behavior, which revealed that the graph rises to both the left and the right. We then factored the polynomial to identify its zeros, finding x = 0 and x = 2, both with a multiplicity of 2. Understanding the multiplicity of these zeros helped us predict that the graph would touch the x-axis at these points but not cross it.

By synthesizing this information, we were able to accurately sketch the graph of f(x), which resembles a "W" shape touching the x-axis at x = 0 and x = 2. This detailed analysis highlights the importance of understanding polynomial functions, including their end behavior, zeros, multiplicity, and how these elements combine to shape the graph.

This exploration provides a solid foundation for further study of polynomial functions and their applications in various mathematical and real-world contexts. The techniques and insights discussed here can be applied to analyze other polynomial functions, enabling a deeper understanding of their behavior and properties.