Factorize each of the following expressions:
(a) `2p(a-b)+3q\ (5a-5b)+4r(2b-2a)`
(b)`a b(a^2+b^2-c^2)+b c(a^2+b^2-c^2)-c a(a^2+b^2-c^2)`
(c)`x(x^2+y^2-z^2)+y(-x^2-y^2+z^2)-z\ (x^2+y^2-z^2)`

Factorize each of the following algebraic expressions: x(x+z)-y(y+z)( ii) a^(2)-b^(2)-a-b

Evaluate each of the following algebraic expressions for x=1,\ y=-1, z=2,\ a=-2,\ b=1,\ c=-2: (i) a x+b y+c z (ii) a x^2+b y^2-c z^2 (iii) a x y+b y z+c x y

Expand the following (i) (4a-b+ac)^(2) (ii) (3a-5b-c)^(2) (iii) (-x+2y-3z)^(2)

If(a^(2)+b^(2)+c^(2))(x^(2)+y^(2)+z^(2))=(ax+by+cz)^(2) shewthatx :a=y:b=z:

Prove that the value of each the following determinants is zero: (a^(x)+a^(-x))^(2),(a^(x)-a^(-x))^(2),1(b^(y)+b^(-y))^(2),(b^(y)-+b^(-y))^(2),1(c^(z)+c^(-z))^(2),(c^(z)-c^(-z))^(2),1]|

Without expanding, show that "Delta"=|(a-x)^2(a-y)^2(a-z)^2(b-x)^2(b-y)^2(b-z)^2(c-x)^2(c-y)^2(c-z)^2|=2(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)

Let a,b,c be the real numbers.The following system of equations in x,y, and z(x^(2))/(a^(2))+(y^(2))/(b^(2))-(z^(2))/(a^(2))=1,(x^(2))/(a^(2))-(y^(2))/(b^(2))+(z^(2))/(a^(2))=1,-(x^(2))/(a^(2))+(y^(2))/(b^(2))+(z^(2))/(a^(2))=1 has a.no solution b.unique solution c .infinitely many solutions d.finitely many solutions

If (a^(2)+b^(2)+c^(2))(x^(2)+y^(2)+z^(2))=(ax+by+cz)^(2), then show that x:a=y:b=z:c

If a(y+z)=b(z+x)=c(x+y) then show that (a-b)/(x^(2)-y^(2))=(b-c)/(y^(2)-z^(2))=(c-a)/(z^(2)-x^(2))]