Question

X+y+z=5 andxy+yz+zx=10 then prove that X cube+y cube+z cube-3xyz=-25

Answer 1

Answer:

(-25)

Step-by-step explanation:

x^3 + y^3 + z^3 - 3xyz =

(x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx)

=5{x^2+y^2+z^2-(xy+yz+zx)}

=5(x^2+y^2+z^2-10)

=5(x^2+y^2+z^2)-50

You know that,

(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx

Now,

x^2+y^2+z^2=(x+y+z)^2-(2xy+2yz+2zx)

x^2+y^2+z^2=5^2-2(xy+yz+zx)

x^2+y^2+z^2=25-2(10)

=5

Substituting x^2+y^2+z^2 by 5,

x^3+y^3+z^3-3xyz

=5(5)-50

=25-50

=(-25)