Show that the function `z = 2x^2 + 2xy + y^2-2x + 2y + 2` is not smaller than `-3.`

Show that x - 2y is a factor or 3x^(3) - 2x^(2) y - 13xy^(2) + 10y^(3) .

If x = 6, y = 4 and z = 3, then find the value of x ^(2) + y ^(2) + z ^(2) - 2xy + 2 yz - 2 zx.

What is the LCM of (x^2 - y^2 – z^2 – 2yz),(x^2 - y^2 + z^2 + 2xz) and (x^2 + y^2 - z^2 - 2xy) ?

By using properties of determinants , show that : {:[( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) ]:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

By using properties of determinants , show that : {:[( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) ]:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

By using properties of determinants , show that : {:|( x,x^(2) , yz) ,( y,y^(2) , zx ) ,( z , z^(2) , xy ) |:} =( x-y)(y-z) (z-x) (xy+yz+ zx)

If y is the mean proportional between x and z , show that xy + yz is the mean proportional between x^(2) + y^(2) and y^(2) + z^(2).