Complex Conjugate Root Theorem Explained If 5 + 6i Is A Root

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When dealing with polynomial functions, especially those with real coefficients, a fascinating property emerges concerning complex roots. This property is encapsulated in the Complex Conjugate Root Theorem. This article delves into this theorem, explains its significance, and applies it to solve the given problem: If $5 + 6i$ is a root of the polynomial function $f(x)$, which of the following must also be a root of $f(x)$?

Complex Conjugate Root Theorem: The Key Concept

The Complex Conjugate Root Theorem states that if a polynomial with real coefficients has a complex number $a + bi$ (where a and b are real numbers, and i is the imaginary unit) as a root, then its complex conjugate $a - bi$ must also be a root. This theorem is a cornerstone in understanding the behavior of polynomials with real coefficients and their roots. The underlying reason for this theorem lies in the nature of polynomial equations and how complex numbers interact within them. When you perform algebraic operations on a polynomial with real coefficients, involving a complex root, the imaginary parts will only cancel out if the conjugate is also present as a root. This ensures that the final polynomial has real coefficients, as initially assumed.

Why Does This Theorem Hold?

To truly appreciate the theorem, let's dissect its core principle. Consider a polynomial $f(x)$ with real coefficients. If $z = a + bi$ is a root of $f(x)$, then $f(z) = 0$. Now, let's take the conjugate of both sides of the equation $f(z) = 0$. Since the coefficients of $f(x)$ are real, taking the conjugate of the polynomial essentially means taking the conjugate of the complex root. This leads us to $f(\overline{z}) = 0$, where $\overline{z} = a - bi$ is the complex conjugate of z. This mathematical manipulation demonstrates that if $a + bi$ is a root, then $a - bi$ must also be a root. The beauty of this theorem is its ability to predict the existence of another root based on the knowledge of one complex root. It simplifies the process of finding all roots of a polynomial, especially when dealing with higher-degree polynomials where direct factorization might be challenging. The theorem is not just a mathematical curiosity; it has practical applications in various fields, including engineering and physics, where complex numbers are used to model real-world phenomena.

Applying the Theorem to the Problem

In our problem, we are given that $5 + 6i$ is a root of the polynomial function $f(x)$. According to the Complex Conjugate Root Theorem, the complex conjugate of $5 + 6i$ must also be a root. The complex conjugate of a complex number $a + bi$ is $a - bi$. Therefore, the complex conjugate of $5 + 6i$ is $5 - 6i$. This means that if $5 + 6i$ is a root of $f(x)$, then $5 - 6i$ must also be a root. This understanding allows us to quickly identify the correct answer from the given options. The theorem provides a direct and efficient way to solve the problem, avoiding the need for complex calculations or guesswork. It highlights the power of mathematical theorems in simplifying problem-solving and providing insights into the underlying structure of mathematical concepts.

Solving the Problem: Step-by-Step

To solve the problem, we follow a straightforward approach using the Complex Conjugate Root Theorem:

  1. Identify the Given Root: We are given that $5 + 6i$ is a root of the polynomial function $f(x)$.
  2. Apply the Complex Conjugate Root Theorem: The theorem states that if $a + bi$ is a root, then $a - bi$ is also a root.
  3. Find the Complex Conjugate: The complex conjugate of $5 + 6i$ is $5 - 6i$.
  4. Conclusion: Therefore, $5 - 6i$ must also be a root of $f(x)$.

This step-by-step solution clearly demonstrates how the Complex Conjugate Root Theorem simplifies the problem. Instead of needing to perform complex polynomial division or other algebraic manipulations, we can directly apply the theorem to find the other root. This approach not only saves time but also reduces the chances of making errors. The theorem acts as a powerful tool in our mathematical toolkit, allowing us to efficiently tackle problems involving complex roots of polynomials. The elegance of this method lies in its directness and the deep mathematical principle it embodies.

Detailed Explanation of the Solution

The question states that $5 + 6i$ is a root of the polynomial function $f(x)$. Our task is to determine which of the given options must also be a root. Options are:

A. $-5 - 6i$ B. $5 - 6i$ C. $6 - 5i$ D. $6 + 5i$

We know from the Complex Conjugate Root Theorem that if a polynomial with real coefficients has a complex root $a + bi$, then its conjugate $a - bi$ is also a root. The given root is $5 + 6i$, where a = 5 and b = 6. To find its complex conjugate, we simply change the sign of the imaginary part. So, the complex conjugate of $5 + 6i$ is $5 - 6i$. Comparing this with the given options, we see that option B, $5 - 6i$, matches our result. Therefore, $5 - 6i$ must also be a root of the polynomial function $f(x)$. The other options do not represent the complex conjugate of the given root. Option A, $-5 - 6i$, has both the real and imaginary parts negated, which is not the conjugate. Options C and D, $6 - 5i$ and $6 + 5i$, have the real and imaginary parts swapped, which is also incorrect. The Complex Conjugate Root Theorem provides a specific rule for finding the conjugate, and it's crucial to apply it correctly to avoid errors. This detailed explanation reinforces the importance of understanding and applying the theorem accurately.

Analyzing the Incorrect Options

To further solidify our understanding, let's analyze why the other options are incorrect. This will help clarify the application of the Complex Conjugate Root Theorem and prevent common mistakes.

  • Option A: $-5 - 6i$

    This option negates both the real and imaginary parts of the given root $5 + 6i$. While negation can sometimes play a role in root transformations, it does not produce the complex conjugate. The complex conjugate is formed by only changing the sign of the imaginary part. Negating both parts would be relevant if we were looking for the additive inverse, but that's not what the Complex Conjugate Root Theorem dictates. This option represents a misunderstanding of the theorem's specific requirements.

  • Option C: $6 - 5i$

    This option swaps the real and imaginary parts and changes the sign of what was the real part in the original root. This is not the correct way to find the complex conjugate. The Complex Conjugate Root Theorem requires us to keep the real part the same and only change the sign of the imaginary part. Swapping the parts is a different kind of transformation and does not guarantee another root of the polynomial. This highlights the importance of adhering strictly to the theorem's rule.

  • Option D: $6 + 5i$

    Similar to option C, this option also swaps the real and imaginary parts. Additionally, it keeps the sign of the new imaginary part as positive, which is the opposite of what the Complex Conjugate Root Theorem requires. The correct conjugate should have the same real part and the opposite sign of the imaginary part. This option further illustrates the common mistake of not correctly applying the theorem's principle of only changing the sign of the imaginary part.

By understanding why these options are incorrect, we reinforce our understanding of the Complex Conjugate Root Theorem and its proper application. It's not enough to simply remember the theorem; we must also understand its nuances and how it differs from other transformations of complex numbers.

Conclusion: The Power of the Complex Conjugate Root Theorem

In conclusion, the Complex Conjugate Root Theorem is a powerful tool for solving problems involving polynomial functions with real coefficients and complex roots. It states that if $a + bi$ is a root, then $a - bi$ must also be a root. Applying this theorem to the given problem, we found that if $5 + 6i$ is a root of $f(x)$, then $5 - 6i$ must also be a root. This understanding allows us to efficiently solve the problem and avoid unnecessary calculations. The theorem is not just a mathematical trick; it reflects a fundamental property of polynomials with real coefficients and their roots. Its applications extend beyond simple problem-solving, offering insights into the structure and behavior of polynomials. Mastering this theorem is crucial for anyone studying algebra and beyond, as it provides a key to unlocking the mysteries of complex roots and their role in mathematical functions.

The correct answer is B. $5 - 6i$. This exercise underscores the importance of understanding fundamental theorems in mathematics. The Complex Conjugate Root Theorem provides a direct path to the solution, making complex problems manageable with a clear understanding of the underlying principles. This theorem serves as a cornerstone in algebra, offering a glimpse into the elegant relationships between roots and coefficients of polynomials.