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  4. If (x + 9) = 0 is a factor of |x&3&7 2&x&2 7&6&x| = 0 , then the other factor is:

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Q. If (x + 9) = 0 is a factor of $\begin{vmatrix}x&3&7\\ 2&x&2\\ 7&6&x\end{vmatrix} = 0 $ , then the other factor is:

Determinants

Solution:

Let $A = \begin{vmatrix}x&3&7\\ 2&x&2\\ 7&6&x\end{vmatrix} = 0 $
$\Rightarrow \ x (x^2 - 12) - 3 (2x - 14) + 7 (12 - 7x) = 0 $
$\Rightarrow \ x^3 - 12x - 6x + 42 + 84 - 49x = 0 $
$\Rightarrow \ x^3 - 67x + 126 = 0 $
If (x + 9) is a factor of the given equation then $(x + 9) (x^2 - 9x + 14) = 0 $
$\Rightarrow \ x^2 - 9x + 14 = 0 $
Thus (x - 7) (x - 2) = 0 is the other factor.