Find the GCD of the following $x^{2} + 14 x + 33, x^{3} + 10 x^{2} - 11 x$

x + 11

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x - 11

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x + 11^{2}

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None of these

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Solution

The correct option is A $x + 11$
We first factorize the given polynomials $x^{2} + 14 x + 33$ and $x^{3} + 10 x^{2} - 11 x$ as shown below:

$x^{2} + 14 x + 33 = x^{2} + 3 x + 11 x + 33 = x (x + 3) + 11 (x + 3) = (x + 3) (x + 11)$

$x^{3} + 10 x^{2} - 11 x = x (x^{2} + 10 x - 11) = x (x^{2} - x + 11 x - 11) = x [x (x - 1) + 11 (x - 1)] = x (x - 1) (x + 11)$

The common factor of $x^{2} + 14 x + 33$ and $x^{3} + 10 x^{2} - 11 x$ is $(x + 11)$ , therefore, the GCD is $x + 11$ .

Hence, the greatest common divisor of $x^{2} + 14 x + 33$ and $x^{3} + 10 x^{2} - 11 x$ is $x + 11$ .

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