Divide: `6x^4y z-3x y^3z+8x^2y z^2y z^4\ \ ` by`\ \ 2x y z` `2/3a^2b^2c^2+4/3a b^2c^3-1/5a b^3c^2` b`y\ \ 1/2a b c`

Divide: 6x^3y^2z^2\ \ by \ \ 3x^2y z (2) 15 m^2n^(3\ ) by \ 5m^2n^2 (3) 24 a^3b^3\ by -8a b (4) -21\ a b c^2 by \ 7a b c

Divide: 5x^3-15 x^2+25 x\ b y\ 5x 4z^3+6z^2-z\ b y-1/2z

Multiply : -8/5x^2y z^3\ b y\ -3/4x y^2z (ii) 3/(14)x^2y \ by\ 7/2x^4y

Multiply : 3a b^2c^3\ b y\ 5a^3b^2c (ii) 4x^2y z\ b y-3/2x^2y z^2

x+y+z=3, a^2x+b^2y+c^2z=a^2+b^2+c^2, a^3x+b^3y+c^3z=a^3+b^3+c^3 [a,b,c are unequal and ab+bc+ca ne =0].

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Factorize the following : (1) 9x^2y+3a x y (2) 16 m-4m^2 (3) -4a^2+4a b-4c a (4) x^2y z+x y^2z+x y z^2 (5) a x^2y+b x y^2+c x y z

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f then prove that |x_1y_1z_1x_2y_2z_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)

If a x1 2+b y1 2+c z1 2=a x1 2+b y2 2+c z2 2=a x3 2+b y3 2+c z3 2=d\ a n d\ a x_2x_3+b y_2y_3+c z_2z_3=a x_3x_1+b y_3y_1+c z_3z_1=a x_1x_2+b y_1y_2+c z_1z_2=f then prove that |x_1y_1z_1x_2y_2z_1x_3y_3z_3|=(d-f)[(d+2f)/(a b c)]^(//2)(a , b ,\ c!=0)