The Conjugate Root Theorem states that if the complex number a + ib is a root of a polynomial in one variable with real coefficients, then the complex conjugatea - bi also a root of that polynomial. This is a useful theorem for solving polynomials with real coefficients. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.
Key Concepts
On this page, you should learn to
Understand that complex roots occur in conjugate pairs
The following video will help you understand all the concepts from this page
Finding Roots of a Polynomial Equation
In the following video we will look at an example involving complex roots of a polynomial equation. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.
The conjugate root theorem states that if the complex number \(a+ib\) is a root of a polynomial f(x) in one variable with real coefficients, then the complex conjugate\(a-ib\) also a root of that polynomial.
One root of the equation \(4z^4-4z^3-25z^2+55z-42=0\) is \(1+\frac {\sqrt{3}}{2}i\)
The equation \(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\) has two real roots \(\alpha\) and \(\beta\) and two complex roots \(\gamma\) and \(\delta\) where \(\gamma=2-5i\).
a. Show that \(\alpha+\beta=\frac{1}{2}.\)
b. Find \(\alpha\beta\).
c. Hence find the two real roots α and β.
d. Find the values of p and q.
Hint
a. If γ=2−5i is a root…then δ = 2+5i is also a root. Work out γ+δ and sum of 4 roots
b. Work out γδ and the product of the 4 roots
c. Work out α and β using the two equations for \(\alpha+\beta\) and \(\alpha\beta\).