Explanation of irrational root theorem and imaginary root theorem

The Irrational Root Theorem says if $a + \sqrt{b}$ 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.

$x^3 + x^2 - 3x - 3 = 0$

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

$a^2 + b^2 \in \{ 1, 5\}$

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: $1 = 0 + 1 = 0 + (\pm 1)$ . This means that for this case $a + bi \in \{\pm i \}$ .

Second, for the number 5: $5 = 1 + 4$ and $5 = 4 + 1$ . For this case $a + bi \in \{ \pm1\pm2i, \pm2i \pm1\}$ .

Now we have 9 possible solutions:

$a + bi \in \{ \pm1\pm2i, \pm2i \pm1\}$

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

We’ll use $k = - 1$ . $(a - (- 1))^2 + b^2$ must divide $f(- 1)$ .

$a + 1 + b^2$ must be divisible by 2. We’ll use this test for every possible solution. In the end we are left with:

$a + bi \in \{ \pmi, - 2 \pm i \}$

$x_1 = - 2 + i$ , $x_2 = - 2 - i$

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:

$x_3 = \frac{1}{2} (- 1 + i\sqrt{3} )$ , $x_4 = \frac{1}{2} (- 1 - i\sqrt{3} )$

Imaginary root theorem

As an addition to this theorem, for every whole number $k, (a - k)^2 + b^2$ divides $f(k)$ .

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

$x^4 + 5x^3 + 10x^2 + 9x + 5 = 0$

If this equation has imaginary roots, by the Imaginary Root Theorem, $a^2 + b^2$ must divide 5.

$a^2 + b^2 \in \{ 1, 5 \}$

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: $1 = 0 + 1 = 0 + (\pm 1)^2$ . This means that for this case $a + bi \in \{ \pm 1 \}$ .

Second, for the number 5: $5 = 1 + 4$ and $5 = 4 + 1$ . For this case $a + bi \in \{ \pm 1 \pm 2i, \pm 2i \pm 1 \}$

Now we have 9 possible solutions:

$a + bi \in \{ \pm i, \pm 1 \pm 2i, \pm 2i \pm 1 \}$

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

We’ll use $k = - 1$ . $(a - (- 1))^2 + b^2$ must divide $f(- 1)$ .

$a + 1 + b^2$ must be divisible by 2. We’ll use this test for every possible solution. In the end we are left with:

$a + bi \in \{ \pm i, - 2 \pm i \}$

$x_1 = -2 + i$ , $x_2 = - 2 - i$

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:

$x_3 = \frac{1}{2}(- 1 + i\sqrt{3})$ , $x_4 = \frac{1}{2}(- 1 - i\sqrt{3})$

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