To find all the roots of $x^{12} - 1 = 0$ and identify the roots which are also roots of $x^4 + x^2 + 1 = 0$, we can follow these steps:

Step 1: Find the roots of $x^{12} - 1 = 0$

The equation $x^{12} - 1 = 0$ can be factored using the difference of squares and the fact that it represents the 12th roots of unity.

$x^{12} - 1 = (x^6 - 1)(x^6 + 1)$

We can further factor $x^6 - 1$:

$x^6 - 1 = (x^3 - 1)(x^3 + 1)$

And $x^3 - 1$ and $x^3 + 1$ can be factored as:

$x^3 - 1 = (x - 1)(x^2 + x + 1)$ $x^3 + 1 = (x + 1)(x^2 - x + 1)$

So,

$x^{12} - 1 = (x - 1)(x + 1)(x^2 + x + 1)(x^2 - x + 1)(x^6 + 1)$

Step 2: Find the roots of each factor

• The roots of $x - 1 = 0$ are $x = 1$.
• The roots of $x + 1 = 0$ are $x = -1$.
• The roots of $x^2 + x + 1 = 0$ are the non-real cube roots of unity: $e^{2\pi i / 3}$ and $e^{4\pi i / 3}$.
• The roots of $x^2 - x + 1 = 0$ are the non-real cube roots of unity: $e^{\pi i / 3}$ and $e^{5\pi i / 3}$.
• The roots of $x^6 + 1 = 0$ are the sixth roots of unity: $e^{\pi i / 6}, e^{5\pi i / 6}, e^{7\pi i / 6}, e^{11\pi i / 6}, e^{3\pi i / 2}, e^{\pi i / 2}$.

Step 3: Find the roots of $x^4 + x^2 + 1 = 0$

The equation $x^4 + x^2 + 1 = 0$ can be rewritten as:

$(x^2)^2 + x^2 + 1 = 0$

Let $y = x^2$. Then,

$y^2 + y + 1 = 0$

The roots of this quadratic equation are:

$y = e^{2\pi i / 3}, y = e^{4\pi i / 3}$

Therefore, the roots $x$ are:

$x = \pm \sqrt{e^{2\pi i / 3}}, x = \pm \sqrt{e^{4\pi i / 3}}$

Step 4: Identify common roots

We need to identify which of these roots are common between $x^{12} - 1 = 0$ and $x^4 + x^2 + 1 = 0$.

The roots of $x^4 + x^2 + 1 = 0$ are the cube roots of unity:

$e^{2\pi i / 3}, e^{4\pi i / 3}$

Since these are also roots of unity, they are also solutions to $x^{12} - 1 = 0$.

Conclusion

The roots of $x^{12} - 1 = 0$ include:

$x = 1, -1, e^{2\pi i / 3}, e^{4\pi i / 3}, e^{\pi i / 3}, e^{5\pi i / 3}, e^{\pi i / 6}, e^{5\pi i / 6}, e^{7\pi i / 6}, e^{11\pi i / 6}, e^{3\pi i / 2}, e^{\pi i / 2}$

The roots of $x^4 + x^2 + 1 = 0$ are:

$e^{2\pi i / 3}, e^{4\pi i / 3}$

Therefore, the common roots are $e^{2\pi i / 3}$ and $e^{4\pi i / 3}$.

Would you like any additional details or have any questions?

Here are some more questions you might be interested in:

1. How do we find the 12th roots of unity?
2. What is the geometric representation of roots of unity?
3. How can we solve polynomial equations using factorization?
4. What are the properties of roots of unity?
5. Can you explain the quadratic formula used for finding roots?

Tip

Always check for common roots by comparing the roots of each polynomial equation. This can be done by solving each polynomial separately and then identifying any overlapping roots.