Find the GCD of the following pairs of polynomials $x^{3} - 3 x^{2} + 4 x - 12, x^{4} + x^{3} + 4 x^{2} + 4 x$

x^{2} + 4

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x^{2} - 4

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x^{2} + 1

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None of these

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Solution

The correct option is A $x^{2} + 4$
Let $x^{3} - 3 x^{2} + 4 x - 12 . . . . (1)$

The second polynomial $x^{4} + x^{3} + 4 x^{2} + 4 x$ can be rewritten as:

$x^{4} + x^{3} + 4 x^{2} + 4 x = x (x^{3} + x^{2} + 4 x + 4)$

Let $x^{3} + x^{2} + 4 x + 4 . . . . (2)$

Subtract equation 1 from equation 2 as follows:

$(x^{3} + x^{2} + 4 x + 4) - (x^{3} - 3 x^{2} + 4 x - 12) = 4 x^{2} + 16 = 4 (x^{2} + 4)$

Hence, the GCD of $x^{3} - 3 x^{2} + 4 x - 12$ and $x^{4} + x^{3} + 4 x^{2} + 4 x$ is $(x^{2} + 4)$ .

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