If The Polynomial Function F(x) Has Roots -8, 1, And 6i, What Other Root Must F(x) Have?

When delving into the realm of polynomial functions, understanding the nature and behavior of their roots is paramount. The roots of a polynomial, also known as zeros or solutions, are the values of x that make the polynomial equal to zero. These roots provide critical insights into the function's graph, its factors, and its overall characteristics. One fundamental concept in this area is the Complex Conjugate Root Theorem, which plays a crucial role in identifying potential roots of polynomials, especially when complex numbers are involved.

The Complex Conjugate Root Theorem

The Complex Conjugate Root Theorem is a cornerstone principle in algebra, particularly when dealing with polynomials that have real coefficients. This theorem states that if a polynomial with real coefficients has a complex number as a root, then its complex conjugate must also be a root. In simpler terms, complex roots of polynomials with real coefficients always come in conjugate pairs. A complex number is expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. The complex conjugate of a + bi is a - bi. The theorem implies that if a + bi is a root of a polynomial, then a - bi must also be a root.

Consider a polynomial f(x) with real coefficients. If z = a + bi is a root of f(x), meaning f(z) = 0, then the Complex Conjugate Root Theorem tells us that z̄ = a - bi is also a root of f(x). This theorem is vital because it ensures that if we encounter a complex root, we automatically know there is another root that is its conjugate. For example, if 2 + 3i is a root of a polynomial with real coefficients, then 2 - 3i must also be a root. This pairing of complex roots is not merely a mathematical curiosity; it has profound implications for the structure and factorization of polynomials.

This theorem is particularly useful in constructing and analyzing polynomials. For instance, if you are given a set of roots, including complex roots, you can use the Complex Conjugate Root Theorem to ensure you have a complete set of roots before constructing the polynomial. It also helps in factoring polynomials. If you know one complex root, you know its conjugate is also a root, which allows you to factor the polynomial further. Understanding the Complex Conjugate Root Theorem is thus essential for anyone studying polynomial functions, as it provides a powerful tool for identifying roots and understanding the behavior of these functions.

Application of the Theorem to the Given Problem

In our specific problem, we are given that the polynomial function f(x) has roots -8, 1, and 6i. To determine what else must be a root of f(x), we need to apply the Complex Conjugate Root Theorem. We know that -8 and 1 are real roots, and they do not have any imaginary parts. The root 6i is a purely imaginary number, which can be written in the form 0 + 6i. According to the Complex Conjugate Root Theorem, if a polynomial with real coefficients has a complex root, then its complex conjugate must also be a root. Therefore, the complex conjugate of 6i (or 0 + 6i) is 0 - 6i, which simplifies to -6i.

Thus, if 6i is a root of the polynomial f(x), then -6i must also be a root. This is because complex roots occur in conjugate pairs when the polynomial has real coefficients. The conjugate of 6i is obtained by changing the sign of the imaginary part, which gives us -6i. This understanding allows us to confidently identify the additional root based on the given information and the theorem.

The Complex Conjugate Root Theorem is not just a theoretical concept; it is a practical tool for solving problems related to polynomial functions. In this case, it helps us quickly determine that if 6i is a root, then -6i must also be a root. This principle is essential for anyone studying algebra and polynomial functions, as it provides a straightforward method for identifying all roots of a polynomial, especially when complex numbers are involved.

By applying this theorem, we can eliminate other options. The options provided are:

• A. -6
• B. -6i
• C. 6-i
• D. 6

We have already determined that the conjugate of 6i is -6i. Therefore, the correct answer is B. -6i. The other options do not represent the complex conjugate of 6i and thus cannot be roots based on the Complex Conjugate Root Theorem.

Why the Other Options Are Incorrect

To further solidify our understanding of polynomial roots and the Complex Conjugate Root Theorem, it's crucial to examine why the other options provided are incorrect. This involves a careful analysis of each option in the context of the given roots and the properties of complex conjugates.

Option A suggests -6 as a potential root. However, the Complex Conjugate Root Theorem applies specifically to complex roots, not real numbers. The number -6 is a real number, and there is no direct relationship between the real root -8 and a possible root of -6 based on the information provided. Real roots do not necessitate the presence of any specific additional real roots in the same way that complex roots necessitate their conjugates. Therefore, -6 cannot be confirmed as a root based on the given roots.

Option C proposes 6 - i as a root. While this is a complex number, it is not the conjugate of the given complex root 6i. The complex conjugate of 6i (or 0 + 6i) is 0 - 6i, which simplifies to -6i. The number 6 - i has a real part of 6 and an imaginary part of -1, which does not correspond to the conjugate of 6i. Therefore, 6 - i is not a necessary root based on the Complex Conjugate Root Theorem.

Option D suggests 6 as a potential root. Similar to option A, 6 is a real number, and its presence as a root is not dictated by the complex root 6i. The Complex Conjugate Root Theorem only guarantees that if a complex number is a root, its conjugate must also be a root. It does not provide information about additional real roots. Thus, 6 cannot be confirmed as a root based on the given information.

In summary, the Complex Conjugate Root Theorem is specific to complex roots and their conjugates. It does not apply to real roots or to complex numbers that are not conjugates of the given complex roots. This distinction is crucial for accurately identifying the roots of polynomial functions. The conjugate of 6i is -6i, making option B the only correct answer.

Conclusion

In conclusion, the question of what else must be a root of the polynomial function f(x), given the roots -8, 1, and 6i, highlights the importance of the Complex Conjugate Root Theorem. This theorem ensures that if a polynomial with real coefficients has a complex root, its conjugate must also be a root. Applying this principle, we determined that the conjugate of 6i is -6i, making option B, -6i, the correct answer. The other options were incorrect because they did not represent the complex conjugate of 6i or because the theorem does not apply to real roots in the same way.

Understanding the Complex Conjugate Root Theorem is essential for solving problems related to polynomial functions, especially those involving complex roots. It provides a clear and direct method for identifying additional roots based on the given information. This theorem is not only a fundamental concept in algebra but also a practical tool for constructing and analyzing polynomials. By mastering this concept, students and practitioners can confidently address a wide range of problems involving polynomial roots.

This exploration of the Complex Conjugate Root Theorem underscores its significance in the broader context of polynomial theory. The ability to quickly identify and apply this theorem can significantly enhance problem-solving skills and deepen understanding of the behavior of polynomial functions. Whether in academic settings or practical applications, the Complex Conjugate Root Theorem remains a valuable asset in the study of algebra and mathematics.