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  4. If x = -9 is a root of A = | x 3 7 2 x 2 7 6 x | = 0, then other two root are

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Q. If x = -9 is a root of A = $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \\ \end{vmatrix}$ = 0, then other two root are

VITEEEVITEEE 2006

Solution:

GivenA = $\begin{vmatrix} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \\ \end{vmatrix}$ = 0
$\Rightarrow $ x[x 2 - 12] - 3[2x- 14] + 7[12 - 7x] = 0
$\Rightarrow \ x^3$ - 67x + 126 = 0
But given (x = 9) is a root of given determinant
$\therefore \ \ (x+9)$is a factor
$\Rightarrow \ \ x^3+9x^2-9x^2 -$81x + 14x + 126 = 0
$\Rightarrow \ \ \ x^2$(x + 9) - 9x(x + 9) + 14(x + 9) = 0
$\Rightarrow \ \ (x \ + \ 9) (x^2$-9x +14)=0
$\Rightarrow \ \ (x \ + \ 9) (x^2$-7x -2x+14)=0
$\Rightarrow $ (x + 9) (x - 7) (x - 2) = 0
$\Rightarrow \ \ x \ = \ -9,7,2$