Find the L.C.M. and H.C.F. of $x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1, x^{3} - 1, x^{3} + x^{2} + x$ , and $x^{2} y + y x + y$ .

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Solution

$x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1 = (x^{2} + x + 1) (x^{2} + x + 1)$
$x^{3} - 1 = (x - 1) (x^{2} + x + 1)$
$x^{3} + x^{2} + x = x (x^{2} + x + 1)$
$x^{2} y + y x + y = y (x^{2} + x + 1)$
L.C.M. $= (x^{2} + x + 1)^{2} (x - 1) x y$
and, H.C.F. $= x^{2} + x + 1.$

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