find the L.C.M. of the following algebraic expressions: $x^{2} - y^{2} + z^{2} - 2 x z, x^{2} - y^{2} - z^{2} + 2 y z, x y + z x + y^{2} - z^{2}$

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Solution

Factors of $x^{2} - y^{2} + z^{2} - 2 x z$
Factors of $x^{2} - y^{2} - z^{2} + 2 y z$
Factors of $x y + z x + y^{2} - z^{2}$
LCM is the lowest common multiple, which is the product of all the common factors taken once and the remaining factors as it is. $∴$ the LCM of $x^{2} - y^{2} + z^{2} - 2 x z, x^{2} - y^{2} - z^{2} + 2 y z, x y + z x + y^{2} - z^{2} ∣ s$
(x-z+y)(x-z-y)(x-y+z)(y+z)

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Show that $x^{2} + x y + y^{2}, z^{2} + z x + x^{2}$ and $y^{2} + y z + z^{2}$ are consecutive terms of an A.P., if $x, y$ and $z$ are in A.P.

x^{2} + x y + y^{2} = 37, y^{2} + y z + z^{2} = 61, z^{2} + z x + x^{2} = 49.

\frac{x^{2} + y^{2} - z^{2} + 2 x y}{x^{2} - y^{2} - z^{2} - 2 y z} .

x^{2} + y^{2} + z^{2}

x^{2} + x y + x z = 135, y^{2} + y z + y x = 351

z^{2} + z x + z y = 243

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