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Question
Solve the following determinant equation:
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Solution
\[\text{ Let }∆ = \begin{vmatrix}1 & x & x^3 \\ 1 & b & b^3 \\ 1 & c & c^3\end{vmatrix}\]
\[ = \begin{vmatrix}1 & x & x^3 \\ 0 & b - x & b^3 - x^3 \\ 1 & c & c^3\end{vmatrix} \left[\text{ Applying }R_2 \text{ to }R_2 - R_1 \right]\]
\[ = \begin{vmatrix}1 & x & x^3 \\ 0 & b - x & b^3 - x^3 \\ 0 & c - x & c^3 - x^3\end{vmatrix} \left[\text{ Applying }R_3 \text{ to }R_3 - R_1 \right]\]
\[ = \begin{vmatrix}1 & x & x^3 \\ 0 & x - b & x^3 - b^3 \\ 0 & x - c & x^3 - c^3\end{vmatrix}\]
\[ = \left( x - b \right)\left( x - c \right)\begin{vmatrix}1 & x & x^2 \\ 0 & 1 & x^2 + xb + b^2 \\ 0 & 1 & x^2 + xc + c^2\end{vmatrix} \]
\[ ∆ = \left( x - b \right)\left( x - c \right)\left( x\left( c - b \right) - b^2 + c^2 \right) = 0\]
\[x = b, c, - \left( b + c \right)\]
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