Let $\alpha$ be a solution of $x^2 + x + 1 = 0$, and for some a and b in $\mathbb{R}$, $[4 \; a \; b] \begin{bmatrix}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{bmatrix} = [0 \; 0 \; 0]$. If $\frac{4}{\alpha^4} + \frac{m}{\alpha} + \frac{n}{\alpha^2} = 3$, then m + n is equal to

Step 1: Analyze the quadratic equation

The given quadratic equation is $x^2 + x + 1 = 0$. The roots of this equation are the complex cube roots of unity, denoted by $\omega$ and $\omega^2$. Since $\alpha$ is a solution, we can say $\alpha = \omega$. We know that $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.

Step 2: Simplify the expression involving alpha

Substitute $\alpha = \omega$ into the given expression. Since $\omega^3 = 1$, we have $\omega^4 = \omega^3 \cdot \omega = 1 \cdot \omega = \omega$. Also, $\frac{1}{\omega} = \frac{\omega^2}{\omega^3} = \omega^2$ and $\frac{1}{\omega^2} = \frac{\omega}{\omega^3} = \omega$. Substitute these simplified terms back into the expression.

Step 3: Solve for m and n using properties of roots of unity

From $1 + \omega + \omega^2 = 0$, we know that $\omega^2 = -1 - \omega$. Substitute this into the equation. Since $1, \omega, \omega^2$ are linearly independent over real numbers, for an equation of the form $A + B\omega = 0$ where A and B are real, we must have $A=0$ and $B=0$. Here, we have $(-4-m-3) + (n-4-m)\omega = 0$.

Step 4: Formulate and solve system of equations

Equating the real and imaginary parts to zero, we get two linear equations. Solving the first equation for $m$ gives $m = -7$. Substituting this value into the second equation allows us to solve for $n$, yielding $n = -3$.

Step 5: Calculate m + n

Finally, we need to find the sum $m+n$. Substitute the values of $m$ and $n$ we found.

More from this chapter

• Express (1 + i)^4 using De Moivre's theorem.
• Simplify (1 + ²)^5.
• Simplify (1-)/(1- ²).
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• Find all real solutions to the equation: √(x + 3 - 4√(x - 1)) + √(x + 8 - 6√(x - 1)) = 1.
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• MODIFIED: Given that variable_91, variable_91, variable_91 are roots of x³ - 3x² + 4 = 0, find the value of variable_91⁴ + variable_91⁴ + variable_91⁴.

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