Using properties of determinants, prove the following: `|1xx^2x^2 1xxx^2 1|=(1-x^3)^2`

Using properties of determinants, prove the following: |[1,x,x^2],[x^2, 1,x],[x,x^2,1]|=(1-x^3)^2

Using properties of determinants, prove the following: |[x,x^2,1+px^3],[y,y^2,1+py^3],[z,z^2,1+pz^3]|=(1+pxyz)(x-y)(y-z)(z-x)

Using properties of determinants,prove the following: det[[1,a,a^(2)a^(2),1,aa,a^(2),1]]=(1-a^(3))^(2)

By using properties of determinants.Show that: det[[1,x,x^(2)x^(2),1,xx,x^(2),1]]=(1-x^(3))^(2)

Using properties of determinants,prove the following: det[[a^(2),ab,acab,b^(2)+1,bcca,cb,c^(2)+1]]=1+a^(2)+b^(2)+c^(2)

Using properties of determinants,prove the following det[[a^(2),ab,acab,b^(2)+1,bcca,cb,c^(2)+1]]=1+a^(2)+b^(2)+c^(2)

Using properties of determinants,show that |1aa^(2)-bc1^(-2)-ca1^(^^2)-ab|=0

Using properties of determinants.Prove that |xx^(2)1+px^(3)yy^(2)1+py^(3)zz^(2)1+pz^(3)|=(1+pxyz)(x-y)(y-z)(z-x) where p is any scalar.

Using properties of determinant prove that: |[1,x+y, x^2+y^2],[1, y+z, y^2+z^2],[1, z+x, z^2+x^2]|= (x-y)(y-z)(z-x)

If f(x)=|a-10axa-1ax^(2)axa|, using properties of determinants,find the value of f(2x)-f(x)