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 conjugate a - 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.
On this page, you should learn to
- Understand that complex roots occur in conjugate pairs
- Solve polynomial equations
- Make links with this topic and Factor and Remainder Theorem and Sums and Products of Roots
The following video will help you understand all the concepts from this page
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 \(1+\frac {\sqrt{3}}{2}i\)
Find the other roots of the equation.
Notes from the video
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