Polynomial Roots Understanding The Complex Conjugate Root Theorem
Polynomial functions are foundational in mathematics, underpinning various concepts across algebra and calculus. When examining polynomial functions, understanding the nature and behavior of their roots is paramount. Roots, also known as zeros or solutions, are the values of x for which the polynomial function f(x) equals zero. These roots offer critical insights into the function's graph, behavior, and overall characteristics. Specifically, the complex conjugate root theorem plays a pivotal role 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 is also a root. This principle is essential for solving polynomial equations and understanding their solutions comprehensively.
The complex conjugate root theorem stems from the properties of complex numbers and their behavior within polynomial equations. A complex number is of 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 asserts that if a + bi is a root of a polynomial f(x) with real coefficients, then a - bi must also be a root. This holds true because when a polynomial with real coefficients is evaluated at a complex number, the imaginary parts must cancel out to yield a real result (zero, in the case of a root). This cancellation occurs precisely because complex conjugates have equal real parts and opposite imaginary parts, ensuring that their contributions to the polynomial's imaginary component negate each other.
Consider a polynomial equation f(x) = 0, where f(x) has real coefficients. If one of the roots is a complex number, say 1 + 2i, then the complex conjugate root theorem dictates that its conjugate, 1 - 2i, must also be a root. This arises from the fundamental operations performed when substituting a complex number into a polynomial. The powers of i cycle through i, -1, -i, and 1, and the real coefficients ensure that the imaginary terms only cancel out if both a complex number and its conjugate are present as roots. This pairing is not merely a coincidence but a mathematical necessity for polynomials with real coefficients. Understanding and applying the complex conjugate root theorem significantly streamlines the process of finding roots, especially for higher-degree polynomials where direct factorization or other methods might be cumbersome. For example, if a quartic polynomial (degree 4) with real coefficients is known to have complex roots 2 + i and 1 - 3i, we immediately know that 2 - i and 1 + 3i are also roots. This knowledge allows us to construct the polynomial or solve related problems more efficiently.
To determine the truth of the given statements about the polynomial function f(x), we must carefully apply the complex conjugate root theorem and related principles. Let's dissect each statement to reveal the underlying mathematics and provide a definitive answer.
H3: Evaluating Statement A
Statement A posits that if 1 + √13 is a root of f(x), then -1 - √13 is also a root. To assess this claim, we must consider the nature of the root 1 + √13. This number is a real number since √13 is a real number (approximately 3.606) and adding 1 to it still results in a real number. The complex conjugate root theorem primarily applies to complex roots, which involve an imaginary component. For real roots, the conjugate is simply the number itself. Therefore, if 1 + √13 is a root, its “conjugate” is also 1 + √13, not -1 - √13. However, another theorem, often referred to as the irrational conjugate theorem, comes into play here. This theorem states that if a polynomial with rational coefficients has a root of the form a + √b, where a and b are rational and √b is irrational, then a - √b is also a root. In our case, 1 + √13 fits this form, where a = 1 and b = 13. Thus, if f(x) has rational coefficients, then 1 - √13 must also be a root. However, the statement suggests that -1 - √13 must be a root, which is not a direct consequence of any theorem. Therefore, Statement A is not necessarily true.
Consider a scenario where the polynomial f(x) has a factor of (x - (1 + √13)) but does not necessarily have a factor of (x - (-1 - √13)). For example, a simple quadratic polynomial f(x) = (x - (1 + √13))(x - 2) has 1 + √13 as a root, but -1 - √13 is not a root. This counterexample demonstrates that Statement A cannot be universally true. The irrational conjugate theorem clarifies that for a root in the form a + √b, the conjugate a - √b must also be a root, provided the polynomial has rational coefficients. The given statement incorrectly suggests a different conjugate, -1 - √13, which does not follow from the theorem. To further illustrate this, let’s construct a polynomial that satisfies 1 + √13 as a root but not -1 - √13. We can define f(x) = (x - (1 + √13))(x - 3). Expanding this, we get f(x) = x² - (4 + √13)x + 3(1 + √13). This polynomial has 1 + √13 as a root, but plugging in -1 - √13 does not result in zero, confirming that -1 - √13 is not necessarily a root. In summary, Statement A makes an incorrect assertion by not adhering to the irrational conjugate theorem and proposing an invalid conjugate root. The root 1 + √13 has a conjugate of 1 - √13, and only if f(x) has rational coefficients, the latter must also be a root. The provided counterexample further solidifies that Statement A is not universally true.
H3: Evaluating Statement B
Statement B asserts that if 1 + 13i is a root of f(x), then 1 - 13i is also a root. This statement directly aligns with the complex conjugate root theorem. If f(x) is a polynomial function with real coefficients, and 1 + 13i is a root, then its complex conjugate, which is 1 - 13i, must also be a root. This theorem holds because complex roots of polynomials with real coefficients always occur in conjugate pairs. To understand why this is true, consider a general polynomial with real coefficients. When a complex number a + bi is substituted into the polynomial, the imaginary terms must cancel out for the result to be zero (since the polynomial has real coefficients). This cancellation can only occur if both a + bi and its conjugate a - bi are roots. The complex conjugate root theorem is a cornerstone in the theory of polynomials, particularly when dealing with complex roots. It simplifies the process of finding roots, as knowing one complex root immediately provides another. In the context of the statement, 1 + 13i is a complex number with a real part of 1 and an imaginary part of 13. Its complex conjugate is 1 - 13i, which has the same real part but an opposite imaginary part. According to the theorem, if 1 + 13i makes f(x) equal to zero, then 1 - 13i must also make f(x) equal to zero.
Consider a quadratic polynomial with roots 1 + 13i and 1 - 13i. We can construct this polynomial as follows: f(x) = (x - (1 + 13i))(x - (1 - 13i)). Expanding this expression, we get f(x) = x² - x(1 - 13i) - x(1 + 13i) + (1 + 13i)(1 - 13i). Simplifying further, f(x) = x² - x + 13ix - x - 13ix + (1 - 169i²). Since i² = -1, the last term becomes (1 + 169) = 170. Thus, f(x) = x² - 2x + 170. This polynomial has real coefficients and confirms that if 1 + 13i is a root, 1 - 13i is also a root. This example explicitly demonstrates the complex conjugate root theorem in action. Therefore, Statement B is true under the condition that the polynomial f(x) has real coefficients. The presence of complex roots in pairs is not a mere coincidence but a fundamental property of polynomials with real coefficients. Without this pairing, the imaginary parts would not cancel out, and the polynomial could not equal zero. In summary, Statement B accurately reflects the complex conjugate root theorem, ensuring that if 1 + 13i is a root of a polynomial f(x) with real coefficients, then 1 - 13i must also be a root.
In conclusion, after evaluating the given statements in light of the complex conjugate root theorem and the irrational conjugate theorem, we find that Statement B is true, while Statement A is not necessarily true. The complex conjugate root theorem guarantees that if 1 + 13i is a root of a polynomial f(x) with real coefficients, then its conjugate, 1 - 13i, must also be a root. This theorem is a crucial tool in understanding and solving polynomial equations, especially those involving complex numbers. Statement A, however, incorrectly applies the concept of conjugates. While the irrational conjugate theorem states that if a polynomial with rational coefficients has a root of the form a + √b, then a - √b is also a root, Statement A suggests an incorrect conjugate. Therefore, Statement B is the correct answer, demonstrating a solid understanding of polynomial roots and their properties.
The correct statement is B: If 1 + 13i is a root of f(x), then 1 - 13i is also a root of f(x).
