Find the LCM of the following: $2 x^{3} - 3 x^{2} - 9 x + 5, 2 x^{4} - x^{3} - 10 x^{2} - 11 x + 8$ whose GCD is $2 x - 1$ .

(x^{3} - 5 x - 8) (2 x^{3} - 3 x^{2} - 9 x + 5)

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(x^{3} + 5 x - 8) (2 x^{3} - 3 x^{2} - 9 x + 5)

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(x^{3} - 5 x - 8) (2 x^{3} + 3 x^{2} - 9 x + 5)

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None of these

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Solution

The correct option is A $(x^{3} - 5 x - 8) (2 x^{3} - 3 x^{2} - 9 x + 5)$

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 the GCD of the polynomials $2 x^{3} - 3 x^{2} - 9 x + 5$ and $2 x^{4} - x^{3} - 10 x^{2} - 11 x + 8$ is $(2 x - 1)$ , therefore, we have:

$(2 x^{3} - 3 x^{2} - 9 x + 5) (2 x^{4} - x^{3} - 10 x^{2} - 11 x + 8) = (2 x - 1) \times L C M \Rightarrow L C M = \frac{(2 x^{3} - 3 x^{2} - 9 x + 5) (2 x^{4} - x^{3} - 10 x^{2} - 11 x + 8)}{(2 x - 1)}$

To find the LCM, we have to do the long division as shown in the above image:

On dividing $2 x^{4} - x^{3} - 10 x^{2} - 11 x + 8$ by $(2 x - 1)$ , the quotient is $x^{3} - 5 x - 8$ and the remainder is $0$ , therefore,

$L C M = (2 x^{2} + x - 5) (x^{3} + 8 x^{2} + 4 x - 21)$

Hence, the LCM of $2 x^{3} - 3 x^{2} - 9 x + 5$ and $2 x^{4} - x^{3} - 10 x^{2} - 11 x + 8$ is $(2 x^{2} + x - 5) (x^{3} + 8 x^{2} + 4 x - 21)$ .

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