Polynomial Roots: Find Factors Using Conjugate Theorem

by Alex Johnson 55 views

In the realm of mathematics, particularly when dealing with polynomial functions, understanding the nature and behavior of roots is paramount. Roots, or zeroes, of a polynomial function are the values of x for which the function equals zero. These roots can be real or complex, and their properties often dictate the characteristics of the polynomial. One powerful tool in analyzing polynomial roots is the complex conjugates theorem, which provides insights into the relationships between complex roots. Let's dive into a scenario involving polynomial roots and leverage the complex conjugates theorem to identify another factor of the polynomial. This article will explore the nuances of polynomial functions, their roots, and how the complex conjugates theorem aids in unraveling their structure.

Understanding the Given Roots

We are given that a polynomial function has the following roots: x - 2, x - (1 + √5), and x + 3i. To fully grasp the implications of these roots, let's break them down individually:

  1. x - 2: This root corresponds to x = 2. It is a real root, meaning that the polynomial function intersects the x-axis at x = 2.
  2. x - (1 + √5): This root corresponds to x = 1 + √5. It is also a real root, but it involves an irrational number (√5). The approximate value of this root is x ≈ 3.236.
  3. x + 3i**: This root corresponds to x = -3i. This is a complex root, specifically a purely imaginary root since it has no real part. Complex roots play a crucial role in polynomial behavior, especially when considering the complex conjugates theorem.

The Complex Conjugates Theorem

The complex conjugates theorem states that if a polynomial with real coefficients has a complex root a + bi, then its complex conjugate a - bi is also a root of the polynomial. In simpler terms, complex roots of polynomials with real coefficients always come in conjugate pairs. This theorem is a cornerstone in understanding the structure and behavior of polynomials, particularly those with real coefficients.

For our given polynomial, we have a complex root x = -3i. This can be written as 0 - 3i, where a = 0 and b = -3. According to the complex conjugates theorem, the complex conjugate of -3i is 0 + 3i, which simplifies to 3i. Therefore, if -3i is a root, then 3i must also be a root of the polynomial.

Applying the Conjugate Root

Given that 3i is also a root of the polynomial, we can express this root as a factor. If x = 3i, then x - 3i is a factor of the polynomial. Therefore, based on the complex conjugates theorem, if x + 3i is a factor, then x - 3i must also be a factor.

Furthermore, let’s consider the other given root, x = 1 + √5. Since this root involves an irrational number, we should consider whether a similar conjugate relationship exists for roots involving irrational numbers. If a polynomial has rational coefficients and a root of the form a + √b, where a and b are rational and √b is irrational, then a - √b is also a root. This is sometimes referred to as the irrational conjugate theorem.

In our case, x = 1 + √5 is a root. Thus, its conjugate, x = 1 - √5, must also be a root. Therefore, the factor corresponding to this conjugate root is x - (1 - √5).

Identifying Another Factor

Now, let's revisit the question: According to the complex conjugates theorem (and the irrational conjugate theorem), which of the following is another factor of the polynomial?

We have already established that if x + 3i is a factor (corresponding to the root x = -3i), then x - 3i must also be a factor (corresponding to the root x = 3i). Additionally, if x - (1 + √5) is a factor (corresponding to the root x = 1 + √5), then x - (1 - √5) must also be a factor (corresponding to the root x = 1 - √5).

Looking at the options provided:

  • A. x + 2
  • B. x - (1 - √5)
  • C. x - √5

Option A, x + 2, corresponds to the root x = -2. We have no information to suggest that x = -2 is a root based on the given roots and the conjugate theorems.

Option B, x - (1 - √5), corresponds to the root x = 1 - √5. As discussed earlier, this is the conjugate of the root x = 1 + √5, and thus, x - (1 - √5) is indeed another factor of the polynomial.

Option C, x - √5, corresponds to the root x = √5. We have no direct evidence to suggest that x = √5 is a root based on the given roots and the conjugate theorems.

Therefore, the correct answer is B. x - (1 - √5).

Conclusion

In conclusion, understanding the properties of polynomial roots and theorems such as the complex conjugates theorem and the irrational conjugate theorem is vital for identifying factors of polynomials. Given the roots x - 2, x - (1 + √5), and x + 3i, we successfully identified x - (1 - √5) as another factor of the polynomial by applying these theorems. This exploration highlights the importance of recognizing conjugate pairs when dealing with complex and irrational roots of polynomials with real or rational coefficients. For further reading, you might find valuable insights on Khan Academy's Polynomial Arithmetic