Complex Roots of Polynomials and De Moivre's Theorem

Complex Conjugate Roots

When working with polynomials that have real coefficients, complex roots always come in conjugate pairs, i.e. if $a + bi$ is a root of a polynomial, then $a - bi$ is also a root of the same polynomial.

This is because of the properties of the sum and product of a polynomial's roots, which are covered further in Topic 2. To summarize, if the coefficients are all real, the sum and product of the roots must be real. If there is a complex root, the only way to make the sum and product of all roots positive is to also include its complex conjugate as a root.

For example, if we have the quadratic $x^2 + 1 = 0$:

The roots are $i$ and $-i$ (conjugate pairs)
We can verify: $(x-i)(x+i) = x^2 + 1$

Tip

A polynomial with real coefficients can only have an even number of complex roots, as they always come in conjugate pairs.

De Moivre's Theorem

De Moivre's theorem is a result formalizing the properties of complex numbers when raised to a power. To derive it, we can start with a complex number in Euler form:

$$re^{i\theta}$$

If we raise it to a power $n$, we get:

$$(re^{i\theta})^n$$

We can apply normal exponent rules to expand this into:

$$r^ne^{in\theta}$$

If we then replace $e^{i\theta}$ with $\cos\theta + i\sin\theta$, we get de Moivre's theorem. For any complex number in polar form:

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Complex Numbers: Recap

Before diving into complex roots and De Moivre's Theorem, let's briefly recap complex numbers and their forms:

A complex number is in the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part.
The imaginary unit $i$ satisfies $i^2 = -1$ .
Complex numbers can also be represented in polar form as $r(\cos \theta + i\sin \theta)$ $r (cos θ + i sin θ)$ , where:
- $r = \sqrt{a^2 + b^2}$ is the magnitude
- $\theta = \tan^{-1}(\frac{b}{a})$ is the argument

DefinitionComplex NumberA number in the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

DefinitionPolar FormA way to represent complex numbers as $r(\cos \theta + i\sin \theta)$ or $r\text{ cis }\theta$ , where $r$ is the magnitude and $\theta$ is the argument.

ExampleConvert

3 + 4i

to polar form:

Magnitude: $r = \sqrt{3^2 + 4^2} = 5$
Argument: $\theta = \tan^{-1}(\frac{4}{3}) \approx 0.93$ radians (quadrant 1)
Polar form: $5(\cos 0.93 + i\sin 0.93)$

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers Notes