Find the GCD of the following $x^{2} - x - 2, x^{2} + x - 6, 3 x^{2} - 13 x + 14$

x - 2

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x + 2

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x - 3

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None of these

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Solution

The correct option is A $x - 2$
We first factorize the given polynomials $x^{2} - x - 2, x^{2} + x - 6$ and $3 x^{2} - 13 x + 14$ as shown below:

$x^{2} - x - 2 = x^{2} - 2 x + x - 2 = x (x - 2) + 1 (x - 2) = (x - 2) (x + 1)$

$x^{2} + x - 6 = x^{2} + 3 x - 2 x - 6 = x (x + 3) - 2 (x + 3) = (x - 2) (x + 3)$

$3 x^{2} - 13 x + 14 = 3 x^{2} - 6 x - 7 x + 14 = 3 x (x - 2) - 7 (x - 2) = (x - 2) (3 x - 7)$

The common factor of $x^{2} - x - 2, x^{2} + x - 6$ and $3 x^{2} - 13 x + 14$ is $(x - 2)$ , therefore, the GCD is $x - 2$ .

Hence, the greatest common divisor is $x - 2$ .

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