Find the other polynomial q (x) of the following, given that LCM, GCD and one polynomial p(x) respectively. $(x^{3} - 4 x) (5 x + 1), (5 x^{2} + x), (5 x^{3} - 9 x^{2} - 2 x)$ .

x (x + 2) (5 x + 1)

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x (x - 2) (5 x + 1)

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x (x + 2) (5 x - 1)

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None of these

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Solution

The correct option is A $x (x + 2) (5 x + 1)$

We know that if $p (x)$ and $q (x)$ are two polynomials, then $p (x) \times q (x) =$ {GCD of $p (x)$ and $q (x)$ } $\times$ {LCM of $p (x)$ and $q (x)$ }.

Now, it is given that one of the polynomial is $p (x) = 5 x^{3} - 9 x^{2} - 2 x$ , the LCM is $(x^{3} - 4 x) (5 x + 1)$ and the GCD is $5 x^{2} + x$ , therefore, we have:

$(5 x^{3} - 9 x^{2} - 2 x) (q (x)) = (5 x^{2} + x) \times (x^{3} - 4 x) (5 x + 1)$

$\Rightarrow [x (5 x^{2} - 9 x - 2)] [q (x)] = [x (5 x + 1)] \times [x (x^{2} - 4) (5 x + 1)]$

$\Rightarrow [x (5 x^{2} - 10 x + x - 2)] [q (x)] = x^{2} (5 x + 1)^{2} (x^{2} - 2^{2})$

$\Rightarrow [x (5 x (x - 2) + 1 (x - 2)] [q (x)] = x^{2} (5 x + 1)^{2} (x + 2) (x - 2) (∵ (a + b)^{2} = a^{2} + b^{2} + 2 a b)$

$\Rightarrow [x (5 x + 1) (x - 2)] [q (x)] = x^{2} (5 x + 1)^{2} (x + 2) (x - 2)$

$\Rightarrow q (x) = \frac{x^{2} (5 x + 1)^{2} (x + 2) (x - 2)}{x (5 x + 1) (x - 2)}$

$\Rightarrow q (x) = x (5 x + 1) (x + 2)$

Hence, the other polynomial $q (x)$ is $x (5 x + 1) (x + 2)$ .

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