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Sagot :
Let's solve the equation [tex]\((1+\omega^2)^4 = \omega\)[/tex] step-by-step.
1. Rewrite the equation:
[tex]\[ (1 + \omega^2)^4 - \omega = 0 \][/tex]
2. Introduce a new variable:
Let's denote [tex]\( z = 1 + \omega^2 \)[/tex]. Then the equation becomes:
[tex]\[ z^4 - \omega = 0 \][/tex]
But we also know that [tex]\( z = 1 + \omega^2 \)[/tex], so we have a system of equations:
[tex]\[ z^4 = \omega \][/tex]
[tex]\[ z = 1 + \omega^2 \][/tex]
3. Combine the equations:
Substitute [tex]\( \omega = z^4 \)[/tex] from the first equation into the second:
[tex]\[ z = 1 + (z^4)^2 \][/tex]
Simplifying, we get:
[tex]\[ z = 1 + z^8 \][/tex]
[tex]\[ z^8 + z - 1 = 0 \][/tex]
4. Solve for [tex]\( z \)[/tex]:
Solve the polynomial [tex]\( z^8 + z - 1 = 0 \)[/tex].
5. Find [tex]\( \omega \)[/tex]:
We need to revert back to [tex]\( \omega \)[/tex] using [tex]\( \omega = z^4 \)[/tex].
Solving this polynomial equation is quite complex, and we get a combination of real and complex roots. Let's list the roots for [tex]\( z \)[/tex]:
The roots of this equation would be complex numbers, and specifically, [tex]\(z\)[/tex] would be a complex number such that:
[tex]\[ z \approx -1/2 \pm \sqrt{3}i/2 \][/tex]
These solutions correspond to:
[tex]\[ z = -1/2 - \sqrt{3}i/2 \quad \text{and} \quad z = -1/2 + \sqrt{3}i/2 \][/tex]
Convert these complex roots back to [tex]\( \omega \)[/tex]:
[tex]\[ \omega = z^4 \][/tex]
So we get:
[tex]\[ \omega = \left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)^4 \quad \text{and} \quad \omega = \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)^4 \][/tex]
6. Find the complex roots involving Root of Unity:
Aside from these complex roots, we also notice that for higher-degree polynomials, SymPy provides roots in terms of `CRootOf`, showing the complexity and number of solutions.
To summarize, the solutions to the equation [tex]\((1 + \omega^2)^4 = \omega\)[/tex] are:
[tex]\[ \omega = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \quad -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad \text{and complex roots represented as CRootOf} \][/tex]
Where the remaining solutions are:
[tex]\[ \omega = \text{CRootOf}(x^6 - x^5 + 4x^4 - 3x^3 + 5x^2 - 2x + 1, k) \][/tex]
for [tex]\(k = 0, 1, 2, 3, 4, 5\)[/tex].
1. Rewrite the equation:
[tex]\[ (1 + \omega^2)^4 - \omega = 0 \][/tex]
2. Introduce a new variable:
Let's denote [tex]\( z = 1 + \omega^2 \)[/tex]. Then the equation becomes:
[tex]\[ z^4 - \omega = 0 \][/tex]
But we also know that [tex]\( z = 1 + \omega^2 \)[/tex], so we have a system of equations:
[tex]\[ z^4 = \omega \][/tex]
[tex]\[ z = 1 + \omega^2 \][/tex]
3. Combine the equations:
Substitute [tex]\( \omega = z^4 \)[/tex] from the first equation into the second:
[tex]\[ z = 1 + (z^4)^2 \][/tex]
Simplifying, we get:
[tex]\[ z = 1 + z^8 \][/tex]
[tex]\[ z^8 + z - 1 = 0 \][/tex]
4. Solve for [tex]\( z \)[/tex]:
Solve the polynomial [tex]\( z^8 + z - 1 = 0 \)[/tex].
5. Find [tex]\( \omega \)[/tex]:
We need to revert back to [tex]\( \omega \)[/tex] using [tex]\( \omega = z^4 \)[/tex].
Solving this polynomial equation is quite complex, and we get a combination of real and complex roots. Let's list the roots for [tex]\( z \)[/tex]:
The roots of this equation would be complex numbers, and specifically, [tex]\(z\)[/tex] would be a complex number such that:
[tex]\[ z \approx -1/2 \pm \sqrt{3}i/2 \][/tex]
These solutions correspond to:
[tex]\[ z = -1/2 - \sqrt{3}i/2 \quad \text{and} \quad z = -1/2 + \sqrt{3}i/2 \][/tex]
Convert these complex roots back to [tex]\( \omega \)[/tex]:
[tex]\[ \omega = z^4 \][/tex]
So we get:
[tex]\[ \omega = \left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right)^4 \quad \text{and} \quad \omega = \left(-\frac{1}{2} + \frac{\sqrt{3}}{2}i\right)^4 \][/tex]
6. Find the complex roots involving Root of Unity:
Aside from these complex roots, we also notice that for higher-degree polynomials, SymPy provides roots in terms of `CRootOf`, showing the complexity and number of solutions.
To summarize, the solutions to the equation [tex]\((1 + \omega^2)^4 = \omega\)[/tex] are:
[tex]\[ \omega = -\frac{1}{2} - \frac{\sqrt{3}}{2}i, \quad -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \quad \text{and complex roots represented as CRootOf} \][/tex]
Where the remaining solutions are:
[tex]\[ \omega = \text{CRootOf}(x^6 - x^5 + 4x^4 - 3x^3 + 5x^2 - 2x + 1, k) \][/tex]
for [tex]\(k = 0, 1, 2, 3, 4, 5\)[/tex].
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