VCE Together Year 12 Specialist Mathematics: Conjugate Root Theorem

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Conjugate Root Theorem

Conjugate Root Theorem is a part of the VCE Specialist Maths topic Algebra and subtopic Factoring Complex Numbers. The conjugate root theorem states that:

  • if a + bi is a root of a polynomial in one variable with real coefficients, then the complex conjugate a – bi will also be a root.

A polynomial is an algebraic expression that consists of variables and coefficients, such as:

x^2 + 3x – 4

A root of a polynomial is a value of the variable that makes the polynomial equal to zero. For example, in the case of the polynomial x^2 + 3x – 4, the roots are x = -1 and x = 4.

The Conjugate Root Theorem states that if a polynomial has a complex root, then its conjugate is also a root of the polynomial.

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. A complex number’s conjugate is the number that can be written as a – bi, which is identical to the original complex number but with the opposite sign of the imaginary part.

For example, if the roots of a polynomial are 3 + 4i and 3 – 4i, then the polynomial is of the form (x-3-4i)(x-3+4i) = (x-3)^2 +(-4i)^2 = (x-3)^2 +16.

The Conjugate Root Theorem is a useful tool for solving polynomial equations and for understanding the behaviour of polynomials near their roots. It also relates to the fact that if a polynomial of degree n has real coefficients, then it has n roots, counting multiplicity, and if the roots are complex numbers, then they come in conjugate pairs.

How Do I Apply the Conjugate Root Theorem?

This next video will help you to understand and apply this complex number theorem.

More Practice!

To make it easier to understand, these next two videos provide plenty of practice questions to test your skills.

Part 1

Part 2

Want to learn more? Check out more of our VCE Mathematics resources here!

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