**The Irrational Root Theorem** says if *is also a root of observed polynomial. In other words, irrational roots come in conjugate pairs.*

Example 1. Find the rational and irrational roots of the following polynomial equation.

If this equation has imaginary roots, by the Imaginary Root Theorem, must divide 5.

Now we have to think all the ways these numbers can be written as the sum of two squares of complex numbers.

First, for the number 1: . This means that for this case .

Second, for the number 5: and . For this case .

Now we have 9 possible solutions:

To eliminate few of these solution we’ll use the noted addition to the Imaginary Root Theorem.

We’ll use . must divide .

must be divisible by 2. We’ll use this test for every possible solution. In the end we are left with:

,

After finding these two solutions we have a quadratic equation left to solve. This will lead us to the remaining two solutions to the given equation:

,

**Imaginary root theorem**

As an addition to this theorem, for every whole number divides .

Example 2. Find all the roots of the following polynomial equation:

If this equation has imaginary roots, by the Imaginary Root Theorem, must divide 5.

Now we have to think all the ways these numbers can be written as the sum of two squares of complex numbers.

First, for the number 1: . This means that for this case .

Second, for the number 5: and . For this case

Now we have 9 possible solutions:

To eliminate few of these solution we’ll use the noted addition to the Imaginary Root Theorem.

We’ll use . must divide .

must be divisible by 2. We’ll use this test for every possible solution. In the end we are left with:

,

After finding these two solutions we have a quadratic equation left to solve. This will lead us to the remaining two solutions to the given equation:

,