Find the GCD of the following pairs of polynomials $2 x^{3} + 2 x^{2} + 2 x + 2, 6 x^{3} + 12 x^{2} + 6 x + 12$

2 (x^{2} + 1)

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2 (x^{2} - 1)

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2 (x^{2} + 2)

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None of these

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Solution

The correct option is B $2 (x^{2} + 1)$
The given polynomials $2 x^{3} + 2 x^{2} + 2 x + 2$ and $6 x^{3} + 12 x^{2} + 6 x + 12$ can be rewritten as:

$2 x^{3} + 2 x^{2} + 2 x + 2 = 2 (x^{3} + x^{2} + x + 1) 6 x^{3} + 12 x^{2} + 6 x + 12 = 6 (x^{3} + 2 x^{2} + x + 2)$

Let

$x^{3} + x^{2} + x + 1 . . . . (1)$
$x^{3} + 2 x^{2} + x + 2 . . . . . (2)$

Subtract equation 1 from equation 2 as follows:

$(x^{3} + 2 x^{2} + x + 2) - (x^{3} + x^{2} + x + 1) = x^{2} + 1$

And the common factor of $2$ and $6$ is $2$ .

Hence, the GCD of $2 x^{3} + 2 x^{2} + 2 x + 2$ and $6 x^{3} + 12 x^{2} + 6 x + 12$ is $2 (x^{2} + 1)$ .

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The factors of $4 x^{3} + 12 x^{2} + 6 x + 18$ are:

For each pair of polynomials p(x) and q(x) given below find the degree of p(x) + q(x) and p(x)q(x).

(i)

(ii)

(iii)

(iv)

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