Finding All Zeros Of Polynomial F(x) = X^4 - 45x^2 - 196 Given Zero 2i

This article delves into the process of finding all zeros of a polynomial function, given a complex zero. Specifically, we will address the polynomial f(x) = x^4 - 45x^2 - 196, and determine its zeros knowing that 2i is one of them. This exploration will involve utilizing the properties of complex conjugates, polynomial division, and quadratic equations to systematically uncover all the roots of the given polynomial. Mastering these techniques is crucial for anyone studying algebra and polynomial functions, as it provides a comprehensive approach to solving polynomial equations and understanding their behavior.

Understanding Complex Conjugate Root Theorem

The complex conjugate root theorem is a cornerstone in polynomial algebra, particularly when dealing with polynomials that have real coefficients. This theorem states that if a polynomial equation with real coefficients has a complex number a + bi as a root, then its complex conjugate a - bi is also a root. In simpler terms, complex roots of polynomials with real coefficients always come in conjugate pairs. This is because complex roots arise from irreducible quadratic factors with real coefficients, and these factors necessitate both the complex number and its conjugate as roots to ensure the imaginary parts cancel out when the polynomial is expanded.

In the context of our given polynomial f(x) = x^4 - 45x^2 - 196, we are told that 2i is a zero. Since the coefficients of f(x) are real numbers (1, -45, and -196), we can apply the complex conjugate root theorem. Therefore, the conjugate of 2i, which is -2i, must also be a zero of f(x). This immediately gives us two roots of the quartic polynomial. This theorem significantly simplifies the process of finding roots, as identifying one complex root automatically reveals another, effectively halving the work required to find complex solutions. It is a powerful tool in polynomial algebra, allowing for systematic deduction of roots based on the conjugate pairs principle, which is essential for solving polynomial equations efficiently.

Utilizing Polynomial Division

Once we have identified two zeros of the polynomial, we can employ polynomial division to reduce the degree of the polynomial and simplify the problem. Knowing that 2i and -2i are roots of f(x) = x^4 - 45x^2 - 196, we can construct a quadratic factor corresponding to these roots. This factor is obtained by multiplying (x - 2i) and (x + 2i), which results in x^2 + 4. This quadratic factor represents a portion of the original quartic polynomial and contains the information about the roots 2i and -2i.

Now, we can divide the original polynomial f(x) by the quadratic factor x^2 + 4. This process, often performed using long division or synthetic division, yields a quotient polynomial. The quotient represents the remaining portion of the original polynomial after factoring out the known roots. In this case, dividing x^4 - 45x^2 - 196 by x^2 + 4 will give us another quadratic polynomial. This resulting quadratic is significantly easier to solve than the original quartic, as it involves solving a quadratic equation, a standard algebraic procedure. The process of polynomial division is not just about simplifying the equation; it's about isolating the remaining roots in a more manageable form, making the task of finding all zeros much more efficient and straightforward. By sequentially reducing the polynomial's degree, we break down a complex problem into simpler, solvable parts.

Solving the Resulting Quadratic Equation

After performing polynomial division, we obtain a quotient quadratic, which in this case is x^2 - 49. To find the remaining zeros of the original polynomial, we need to solve this quadratic equation. Setting x^2 - 49 = 0 presents a straightforward quadratic equation that can be solved using several methods. One of the simplest methods is factoring, where we recognize x^2 - 49 as a difference of squares, which factors into (x - 7)(x + 7).

Alternatively, we can isolate x^2 by adding 49 to both sides of the equation, resulting in x^2 = 49. Taking the square root of both sides yields x = ±7. This gives us two real roots: x = 7 and x = -7. These roots, along with the complex roots 2i and -2i that we found earlier, constitute the complete set of zeros for the polynomial f(x) = x^4 - 45x^2 - 196. Solving the resulting quadratic equation is a crucial step, as it uncovers the final pieces of the puzzle, providing a comprehensive understanding of the polynomial's behavior and its intersection points with the x-axis in the complex plane. This step demonstrates the power of combining different algebraic techniques to solve complex problems systematically.

The Complete Set of Zeros

By combining the results from the complex conjugate root theorem, polynomial division, and solving the quadratic equation, we can now present the complete set of zeros for the polynomial f(x) = x^4 - 45x^2 - 196. We initially identified 2i as a zero, and by applying the complex conjugate root theorem, we deduced that -2i is also a zero. Subsequently, through polynomial division, we reduced the quartic polynomial to a quadratic equation, x^2 - 49 = 0, which we solved to find the real roots 7 and -7.

Therefore, the zeros of f(x) are 2i, -2i, 7, and -7. This comprehensive solution illustrates the interconnectedness of different algebraic principles. The complex conjugate root theorem ensures that complex roots are accounted for in pairs, polynomial division simplifies the polynomial to a manageable form, and solving the resulting equation reveals the remaining roots. This process not only finds all the zeros but also provides a deeper understanding of the polynomial's structure and behavior. The final list of zeros represents the complete solution set, providing a full picture of where the polynomial intersects the x-axis in the complex plane.

Conclusion

In conclusion, finding all zeros of a polynomial, especially one with complex roots, is a multi-faceted process that requires a strong understanding of several algebraic concepts. Starting with the complex conjugate root theorem, we can identify pairs of complex roots. Then, polynomial division helps reduce the complexity of the polynomial, allowing us to solve for the remaining roots more easily. Finally, methods for solving quadratic equations, such as factoring or using the square root property, provide the tools to find the last zeros.

In the case of f(x) = x^4 - 45x^2 - 196, given the zero 2i, we systematically applied these techniques to find all its zeros: 2i, -2i, 7, and -7. This exercise demonstrates the power and elegance of algebraic methods in unraveling the structure of polynomial functions. By mastering these techniques, students and professionals alike can confidently tackle complex polynomial problems and gain a deeper appreciation for the interconnectedness of mathematical concepts. The process not only yields solutions but also enhances problem-solving skills and mathematical intuition, making it a valuable asset in various fields of study and application.

Therefore, the correct answer is D: 2i, 7, -7