Find the GCD of the following pairs of polynomials $x^{3} - 9 x^{2} + 23 x - 15, 4 x^{2} - 16 x + 12$

x^{2} - 4 x + 3

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x^{2} + 4 x + 3

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x^{2} - 4 x - 3

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None of these

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Solution

The correct option is D $x^{2} - 4 x + 3$
Let

$x^{3} - 9 x^{2} + 23 x - 15 . . . . (1)$
$4 x^{2} - 16 x + 12 . . . . (2)$

Here higher powers are removed by making them equal . Therefore, multiply equation 2 by $x$ and then divide it by $4$ as shown below:

$x (4 x^{2} - 16 x + 12) = 4 x^{3} - 16 x^{2} + 12 x \frac{1}{4} (4 x^{3} - 16 x^{2} + 12 x) = x^{3} - 4 x^{2} + 3 x . . . . . (3)$

Now, subtract equation 3 from equation 1 as follows:

$(x^{3} - 9 x^{2} + 23 x - 15) - (x^{3} - 4 x^{2} + 3 x) = - 5 x^{2} + 20 x - 15 = - 5 (x^{2} - 4 x + 3)$

Hence, the GCD of $x^{3} - 9 x^{2} + 23 x - 15$ and $4 x^{2} - 16 x + 12$ is $(x^{2} - 4 x + 3)$ .

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