How to Find the Remaining Roots of a Polynomial with Given Complex Roots

In this article, we will guide you through the process of finding the remaining roots of a polynomial when given one of its complex roots. We will start by discussing the fundamental concepts and then walk you through the step-by-step solution. By the end of this guide, you’ll be able to solve similar problems efficiently.

Understanding Polynomials and Complex Roots

A polynomial is an expression consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. When dealing with polynomials that have real coefficients, it is a well-known fact that if a complex number is a root, its conjugate is also a root. This property allows us to find other roots of the polynomial more easily.

Given Polynomial and Known Root

Consider the polynomial equation:

z4 - 2z3 - 2z2 - 10z - 25 0

Given that 1 - 2i is a root, we know that the conjugate 1 2i is also a root.

Step-by-Step Solution

To find the remaining roots, we can follow these steps:

1. Identify Known Roots

The roots we have so far are:

We can now form a quadratic equation using these roots.

2. Form a Quadratic Factor

The two known roots can be used to form a quadratic polynomial:

(z - (1 - 2i))(z - (1 2i))

This simplifies as follows:

(z - 1 2i)(z - 1 - 2i) (z - 1)2 - (2i)2 (z - 1)2 4

Further simplification gives:

(z - 1)2 4 z2 - 2z 1 4 z2 - 2z 5

3. Divide the Original Polynomial by the Quadratic Factor

To find the remaining quadratic factor, we will use polynomial long division of the original polynomial:

z4 - 2z3 - 2z2 - 10z - 25

by z2 - 2z 5.

The first term is z2 since z2(z2 - 2z 5) z4 - 2z3 5z2.

Subtract:

z4 - 2z3 - 2z2 - 10z - 25 - (z4 - 2z3 5z2) 4z2 - 10z - 25

The next term is 4z since 4z(z2 - 2z 5) 4z3 - 8z2 20z.

Subtract again:

4z3 - 10z - 25 - (4z3 - 8z2 20z) 5z2 - 30z - 25

The next term is 5 since 5(z2 - 2z 5) 5z2 - 10z 25.

Subtract:

5z2 - 30z - 25 - (5z2 - 10z 25) 0

Thus the division is exact, and we have:

z4 - 2z3 - 2z2 - 10z - 25 (z2 - 2z 5)(z2 4z 5)

4. Solve the Remaining Quadratic Equation

Now, we need to find the roots of the quadratic equation z2 4z 5 0.

Using the quadratic formula:

z frac{-b pm sqrt{b^2 - 4ac}}{2a}

Substitute a 1, b 4, and c 5:

z frac{-4 pm sqrt{4^2 - 4 cdot 1 cdot 5}}{2 cdot 1} frac{-4 pm sqrt{16 - 20}}{2} frac{-4 pm sqrt{-4}}{2} frac{-4 pm 2i}{2}

This simplifies to:

z -2 pm i

Summary of All Roots

The roots of the polynomial are:

By understanding the properties of complex roots and following a systematic approach, you can solve polynomial equations effectively. This method is crucial for various applications in mathematics and engineering. Familiarize yourself with these techniques to handle polynomial equations more confidently.