How To Find The Roots Of The Polynomial Function: F(x) = (3x^4 + 1)^2?

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To determine the roots of the given polynomial function, which is a fundamental task in algebra and calculus, we will explore the intricacies of the function f(x) = (3x⁴ + 1)². Polynomial roots, also known as zeros, are the values of x for which the function equals zero. These roots provide critical information about the function's behavior, such as where it intersects the x-axis, and are crucial in various applications across mathematics, physics, and engineering. This article delves into a comprehensive approach to finding these roots, combining algebraic techniques with a clear understanding of the polynomial's structure. By meticulously examining the equation and applying relevant methods, we will identify all values of x that satisfy the condition f(x) = 0, thereby gaining a thorough understanding of the function's properties and behavior. This process not only enhances our algebraic skills but also provides valuable insights into the broader applications of polynomial functions in mathematical modeling and problem-solving.

Understanding the Polynomial Function

The journey to find the roots begins with a thorough understanding of the polynomial function itself. Our given function is f(x) = (3x⁴ + 1)². This function is a composite function, meaning it is composed of two functions: an inner function g(x) = 3x⁴ + 1 and an outer function h(u) = u², where u = g(x). Understanding this composition is crucial because the roots of the composite function are inherently linked to the roots of its constituent parts. The inner function, 3x⁴ + 1, is a quartic polynomial (degree 4), which means it can have up to four roots, while the outer function, squaring the entire expression, influences the multiplicity of these roots. Multiplicity refers to the number of times a particular root appears as a solution. For instance, a root with a multiplicity of 2 means that the factor corresponding to that root appears twice in the factored form of the polynomial. Recognizing the structure of the function as a squared term also tells us that any roots we find for the inner function will have a multiplicity of 2 in the overall function f(x). This understanding significantly streamlines our approach to finding the roots, as we can focus primarily on the inner function, knowing that its roots will directly inform the roots of the composite function. Moreover, the squared nature of the function guarantees that f(x) will always be non-negative, providing additional context for interpreting the roots and the function's overall behavior. This initial analysis sets the stage for a more targeted and efficient search for the roots, allowing us to apply appropriate algebraic techniques with greater precision and insight.

Setting up the Equation

The pivotal step in determining the roots of f(x) = (3x⁴ + 1)² involves setting the function equal to zero. This is based on the fundamental concept that roots are the x-values for which the function's output is zero. Thus, we establish the equation (3x⁴ + 1)² = 0. This equation now represents the core challenge we aim to solve. Squaring both sides of the equation is a common algebraic technique used to simplify equations involving squared terms. In this case, squaring both sides effectively removes the outer square in our function, leading to a simpler form that is easier to manipulate. Applying this operation results in the equation 3x⁴ + 1 = 0. This simplified equation is a quartic equation, specifically a quartic polynomial set equal to zero. Although it is a degree 4 polynomial, its form is relatively straightforward, lacking the cubic and quadratic terms, which simplifies the solution process significantly. This simplification is a crucial step because solving general quartic equations can be complex and may require advanced algebraic methods. However, the absence of certain terms allows us to employ more direct techniques, such as isolating the term with x and then taking roots. This strategic simplification transforms the problem into a more manageable form, enabling us to apply targeted algebraic manipulations to find the roots. The act of setting up the equation and simplifying it is a critical phase in the root-finding process, laying the groundwork for the subsequent algebraic steps that will lead us to the solutions.

Solving for x

With the simplified equation 3x⁴ + 1 = 0 in hand, we proceed to solve for x, the heart of finding the polynomial's roots. The first step is to isolate the term containing x. We subtract 1 from both sides of the equation, resulting in 3x⁴ = -1. This manipulation brings us closer to isolating x by moving the constant term to the other side of the equation. Next, we divide both sides by 3 to further isolate the x⁴ term, giving us x⁴ = -1/3. This equation is now in a form where we can directly address the x⁴ term. To solve for x, we need to take the fourth root of both sides. However, the presence of a negative number on the right side, (-1/3), introduces the realm of complex numbers. This is because taking an even root (like the fourth root) of a negative number results in complex solutions. Recall that complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as i² = -1. Taking the fourth root of x⁴ = -1/3 requires us to find complex numbers that, when raised to the fourth power, equal -1/3. This step is crucial because it highlights the importance of considering complex solutions when dealing with polynomial equations, particularly those with even powers and negative constants. The introduction of complex numbers expands the solution space and ensures that we find all possible roots of the polynomial. The process of solving for x in this context not only demonstrates the algebraic manipulations involved but also underscores the fundamental connection between polynomial equations and complex number theory.

Finding the Complex Roots

The task of finding the complex roots of x⁴ = -1/3 requires us to delve into the world of complex numbers and their representation. Recall that complex numbers can be expressed in polar form as r(cos θ + i sin θ), where r is the magnitude (or modulus) of the complex number, and θ is the argument (or angle) it makes with the positive real axis in the complex plane. Expressing -1/3 in polar form is the key to finding its fourth roots. The magnitude of -1/3 is simply 1/3, and its argument is π (or 180 degrees) because it lies on the negative real axis. Therefore, -1/3 can be written as (1/3)(cos π + i sin π) in polar form. Now, we apply De Moivre's Theorem, a fundamental theorem in complex number theory, which states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth roots are given by r^(1/n) [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], where k is an integer ranging from 0 to n - 1. Applying De Moivre's Theorem to our equation, we find the four fourth roots of -1/3 by substituting n = 4, r = 1/3, and θ = π, and letting k take on the values 0, 1, 2, and 3. This process yields four distinct complex roots, each corresponding to a different value of k. These roots are spread evenly around the complex plane, a characteristic feature of the roots of complex numbers. Calculating these roots involves finding the fourth root of 1/3 (which is a real number) and then evaluating the cosine and sine of the angles given by (π + 2πk)/4. The resulting complex numbers are the solutions to the equation x⁴ = -1/3, and hence, are the roots of the polynomial function f(x). The application of De Moivre's Theorem not only provides a method for finding the roots but also illustrates the elegant interplay between algebra, trigonometry, and complex number theory.

Listing the Roots

After applying De Moivre's Theorem, we can now list the roots of the polynomial function f(x) = (3x⁴ + 1)². These roots, as calculated in the previous section, are complex numbers. Let's denote the fourth root of 1/3 as ∜(1/3). The four roots can then be expressed as follows:

  1. x₁ = ∜(1/3) [cos(π/4) + i sin(π/4)]
  2. x₂ = ∜(1/3) [cos(3π/4) + i sin(3π/4)]
  3. x₃ = ∜(1/3) [cos(5π/4) + i sin(5π/4)]
  4. x₄ = ∜(1/3) [cos(7π/4) + i sin(7π/4)]

These roots can be further simplified by evaluating the trigonometric functions. Recall that cos(π/4) = sin(π/4) = √2/2, cos(3π/4) = -√2/2, sin(3π/4) = √2/2, cos(5π/4) = -√2/2, sin(5π/4) = -√2/2, cos(7π/4) = √2/2, and sin(7π/4) = -√2/2. Substituting these values, we get:

  1. x₁ = ∜(1/3) (√2/2 + i √2/2)
  2. x₂ = ∜(1/3) (-√2/2 + i √2/2)
  3. x₃ = ∜(1/3) (-√2/2 - i √2/2)
  4. x₄ = ∜(1/3) (√2/2 - i √2/2)

It's important to remember that since the original function was f(x) = (3x⁴ + 1)², the roots have a multiplicity of 2. This means that each of the roots listed above appears twice as a solution. The concept of multiplicity is crucial in polynomial theory because it affects the behavior of the function near its roots. For instance, a root with a multiplicity of 2 indicates that the function touches the x-axis at that point but does not cross it. In summary, the roots of the polynomial function f(x) = (3x⁴ + 1)² are four complex numbers, each with a multiplicity of 2, as listed above. These roots provide a complete understanding of where the function equals zero and contribute to a broader understanding of its properties and behavior in the complex plane.

Conclusion

In conclusion, determining the roots of the polynomial function f(x) = (3x⁴ + 1)² involves a systematic approach that combines algebraic manipulation with complex number theory. We began by understanding the structure of the function, recognizing it as a composite function with an inner quartic polynomial and an outer square. Setting the function equal to zero and simplifying the resulting equation led us to the crucial step of solving for x in the equation x⁴ = -1/3. This equation necessitated the use of complex numbers, as we were taking an even root of a negative number. Applying De Moivre's Theorem, a cornerstone of complex number theory, allowed us to find the four complex fourth roots of -1/3. These roots were then expressed in both polar and rectangular forms, providing a clear representation of their location in the complex plane. Finally, we listed the roots, emphasizing their multiplicity of 2 due to the squared nature of the original function. This multiplicity is a key characteristic that influences the function's behavior near its roots. The process of finding these roots not only demonstrates the practical application of algebraic techniques and complex number theory but also highlights the interconnectedness of different areas of mathematics. Understanding the roots of a polynomial function is essential for various applications in mathematics, physics, engineering, and computer science. It allows us to analyze the function's behavior, predict its values, and solve related problems. The exploration of this specific polynomial function, f(x) = (3x⁴ + 1)², provides a valuable case study in the broader field of polynomial root-finding, illustrating the power and elegance of mathematical methods in uncovering the fundamental properties of functions.