Find the GCD of the following pairs of polynomials $3 x^{3} + 18 x^{2} + 33 x + 18, 3 x^{2} + 13 x + 10$

x + 1

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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 $x + 1$
Let

$3 x^{3} + 18 x^{2} + 33 x + 18 . . . . (1)$
$3 x^{2} + 13 x + 10 . . . . (2)$

Here higher powers are removed by making them equal . Therefore, multiply equation 2 by $x$ and then subtract it from equation 1 as shown below:

$x (3 x^{2} + 13 x + 10)$

$= 3 x^{3} + 13 x^{2} + 10 x$ and

$(3 x^{3} + 18 x^{2} + 33 x + 18) - (3 x^{3} + 13 x^{2} + 10 x) = 5 x^{2} + 23 x + 18 . . . . . (3)$

Now, multiply equation 3 by $3$ and equation 2 by $5$ as follows:

$3 (5 x^{2} + 23 x + 18) = 15 x^{2} + 69 x + 54 . . . . . (4) 5 (3 x^{2} + 13 x + 10) = 15 x^{2} + 65 x + 50 . . . . . (5)$

Subtract equation 5 from equation 4:

$(15 x^{2} + 69 x + 54) - (15 x^{2} + 65 x + 50) = 4 x + 4 = 4 (x + 1)$

Hence, the GCD of $3 x^{3} + 18 x^{2} + 33 x + 18$ and $3 x^{2} + 13 x + 10$ is $(x + 1)$ .

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