Simplify each of the following expressions:

Solution:

(i) Expanding, we get

$=\left[x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x\right]+\left[x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}+2 x \frac{y}{2}+2 \frac{z x}{3}+\frac{y z}{3}\right]$

$-\left[\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{10}+\frac{x y}{3}+\frac{y z}{6}+\frac{x z}{4}\right]$

$\left[\because(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 x z\right]$

$=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x+x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}+2 x \frac{y}{2}+\frac{x y}{3}+\frac{2 z x}{3}-\frac{x^{2}}{4}-\frac{y^{2}}{9}-\frac{z^{2}}{10}$

$-\frac{x y}{3}-\frac{y z}{6}-\frac{x z}{4}$

Rearranging coefficients,

$=\frac{8 x^{2}-x^{2}}{4}+\frac{36 y^{2}+9 y^{2}-4 y^{2}}{36}+\frac{144 z^{2}+16 z^{2}-9 z^{2}}{144}+\frac{6 x y+3 x y-x y}{3}+\frac{13 y z}{6}+\frac{29 x z}{12}$

$=\frac{7 x^{2}}{4}+\frac{41 y^{2}}{36}+\frac{151 z^{2}}{144}+\frac{8 x y}{3}+\frac{13 y z}{6}+\frac{29 z x}{12}$

$(x+y+z)^{2}+\left(x+\frac{y}{2}+\frac{z}{3}\right)^{2}-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^{2}$

$=\frac{7 x^{2}}{4}+\frac{41 y^{2}}{36}+\frac{151 z^{2}}{144}+\frac{8 x y}{3}+\frac{13 y z}{6}+\frac{29 z x}{12}$

(ii) Expanding, we get

$(x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y$

$=\left[x^{2}+y^{2}+4 z^{2}+2 x y+2 y(-2 z)+2 a(-2 c)\right]-x^{2}-y^{2}-3 z^{2}+4 x y$

$=z^{2}+6 x y-4 y z-4 z x$

$(x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y=z^{2}+6 x y-4 y z-4 z x$

(iii) Expanding, we get

$\left[x^{2}-x+1\right]^{2}-\left[x^{2}+x+1\right]^{2}$

$\left.=\left(x^{2}\right)^{2}+(-x)^{2}+1^{2}+2\left(x^{2}\right)(-x)+2(-x)(1)+2 x^{2}\right)-\left[\left(x^{2}\right)^{2}+x^{2}+1+2 x^{2} x+2 x(1)+2 x^{2}(1)\right]$

$\left[\because(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 x z\right]$

$=x^{4}+y^{2}+1-2 x^{3}-2 x+2 x^{2}-x^{2}-x^{4}-1-2 x^{3}-2 x-2 x^{2}$

$=-4 x^{3}-4 x$

$=-4 x\left(x^{2}+1\right)$

Hence simplified equation $=\left[x^{2}-x+1\right]^{2}-\left[x^{2}+x+1\right]^{2}=-4 x\left(x^{2}+1\right)$