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LCM of Polynomials

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Question

Find the LCM of $8 x^{3} - y^{3}, a b (4 x^{2} + 2 x y + y^{2}), b c (4 x^{2} - y^{2})$

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Solution

We know that $L C M$ is the least common multiple.

Factorise $8 x^{3} - y^{3}$ as follows:

$8 x^{3} - y^{3} = (2 x)^{3} - y^{3} = (2 x - y) [(2 x)^{2} + y^{2} + (2 x \times y)] = (2 x - y) (4 x^{2} + y^{2} + 2 x y)$
(using identity $a^{3} - b^{3} = (a - b) (a^{2} + b^{2} + a b)$ )

Now, factorise $b c (4 x^{2} - y^{2})$ as follows:

$b c (4 x^{2} - y^{2}) = b c [(2 x)^{2} - y^{2})] = b c (2 x + y) (2 x - y)$ (using identity $a^{2} - b^{2} = (a + b) (a - b)$ )

Therefore, the least common multiple between the polynomials $8 x^{3} - y^{3}$ , $a b (4 x^{2} + 2 x y + y^{2})$ and $b c (4 x^{2} - y^{2})$ is:

$L C M = a b c (2 x - y) (2 x + y) (4 x^{2} + 2 x y + y^{2}) = a b c (2 x + y) (8 x^{3} - y^{3})$
(using identity $a^{3} - b^{3} = (a - b) (a^{2} + b^{2} + a b)$ )

Hence, the $L C M$ is $a b c (2 x + y) (8 x^{3} - y^{3})$ .

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Q. Which of the following is the factorisation of

8 x^{3} + y^{3}

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