Proving the Conjugate Root Theorem for Quadratic Polynomials with Real Coefficients

The conjugate root theorem is a fundamental concept in the study of real polynomials. In this article, we will provide a detailed proof that if a complex number (a bi) is a root of a real quadratic polynomial, then its complex conjugate (a - bi) must also be a root.

Introduction

Real quadratic polynomials play a significant role in various mathematical and engineering applications. The nature of the roots of these polynomials, particularly when they are complex, is governed by the conjugate root theorem. This theorem states that if a polynomial with real coefficients has a complex root, then the complex conjugate of that root must also be a root of the polynomial.

Form of the Quadratic Polynomial

A real quadratic polynomial can be expressed in the standard form:

[P(x) ax^2 bx c]

where (a, b,) and (c) are real coefficients.

Assumption of a Root

Let (r_1 a bi) be a root of the polynomial (P(x)). Given that (P(x)) has real coefficients, we can utilize the property that non-real roots must occur in conjugate pairs. This property stems from the fact that the coefficients of the polynomial are real, meaning that the polynomial can be expressed as the product of linear factors involving both the root and its conjugate.

Conjugate Root Theorem

According to the Conjugate Root Theorem, if (r_1) is a root of a polynomial with real coefficients, then its complex conjugate (r_2 a - bi) must also be a root. This theorem is derived from the fact that the polynomial can be factored into linear factors involving both roots. The proof of this theorem is as follows:

Verification

- We can verify this by substituting (r_2) into the polynomial (P(x)):

[P(a - bi) a(a - bi)^2 b(a - bi) c]

- Simplifying (a - bi^2):

[a - bi^2 a^2 - 2abi - b^2i^2 a^2 - 2abi b^2]

- Thus,

[P(a - bi) a(a^2 - 2abi b^2) b(a - bi) c]

- Simplify the expression

[ a^3 - 2a^2bi ab^2 ba - b^2i c]

- Group real and imaginary parts

[ a^3 ab^2 ba c - 2a^2bi - b^2i]

- Since the polynomial (P(x)) has real coefficients, the imaginary part must be zero. Therefore, the expression will yield zero if (P(a bi) 0), confirming that both roots satisfy the polynomial.

Conclusion

Therefore, since (r_1 a bi) is a root and the coefficients of (P(x)) are real, it follows that (r_2 a - bi) is also a root. This completes the proof. In summary, if (a bi) is a root of a real quadratic polynomial (P(x)), then its complex conjugate (a - bi) must also be a root due to the nature of polynomials with real coefficients.

Mathematical Proof Using Vieta's Formulas

A straightforward and elegant mathematical proof using Vieta's formulas is as follows:

If (x_1 r is in mathbb{R}) is one root of (P(x) ax^2 bx c), it has to fulfill together with the other one (x_2 r' - is' in mathbb{R}). Vieta’s rules: (x_1x_2 -frac{b}{a}) (x_1x_2 frac{c}{a})

Since the right-hand side of the first equation is real, the left-hand side (x_1x_2 rr' - iss') must also be real. Therefore, (s' -s). Additionally, since the right-hand side of the second equation is real, the left-hand side (x_1x_2 rr' - is rr' - s^2) must also be real, implying that (r' r). Hence, the second solution is indeed (x_2 r - is).

Conclusion Summary

The conjugate root theorem is a powerful tool in the analysis of real polynomials. It ensures that if a polynomial with real coefficients has a complex root, then the complex conjugate of that root must also be a root. This property is essential in various mathematical and engineering applications and provides a deeper understanding of the nature of real polynomials.