Complex Numbers - Roots of Polynomials

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.


Key Concepts

On this page, you should learn to

Essentials

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 \(1+\frac {\sqrt{3}}{2}i\)

Find the other roots of the equation.

Notes from the video

Summary

Print from here

Test Yourself

Here is a quiz that practises the skills from this page


START QUIZ!

Exam-style Questions

Question 1

One root of the equation z² + bz + c = 0 is 2+3i where \(b,c\in\mathbb{Z}\).

Find the value of b and the value of c.

Hint

Full Solution

Question 2

\(\frac{2}{1+i}\) is a root to the quadratic equation z² + px + q = 0

a) Find the other root

b) Hence find the values of p and q.

Hint

Full Solution

Question 3

2 - 3i is a root of the equation \(z^3-7z^2+az+b=0\quad ,a, b\in\ \mathbb{R}\)

Work out a and b and the other roots of the equation.

Hint

Full Solution

Question 4

The quartic equation \(z^4+az^3+bz^2+cz+d\) has roots 2 + i and 2i

a) Work out the other roots of the equation

b) Find the values of a , b , c and d

Hint

Full Solution

Question 5

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

Full Solution

MY PROGRESS

How much of Complex Numbers - Roots of Polynomials have you understood?