Find the LCM of $5 (y^{2} - z^{2}), y^{2} + 2 y z + z^{2}$

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Solution

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

Factorise $5 (y^{2} - z^{2})$ as follows:

$5 (y^{2} - z^{2}) = 5 (y - z) (y + z)$ (using identity $a^{2} - b^{2} = (a + b) (a - b)$ )

Now, factorise $y^{2} + 2 y z + z^{2}$ as follows:

$y^{2} + 2 y z + z^{2} = (y + z)^{2}$ (using identity $(a + b)^{2} = a^{2} + b^{2} + 2 a b$ )

Since the least common multiple between the polynomials $5 (y^{2} - z^{2})$ and $y^{2} + 2 y z + z^{2}$ is $5 (y - z) (y + z)^{2}$ .

Hence, the $L C M$ is $5 (y - z) (y + z)^{2}$ .

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