Question

if X + Y + Z is equal to 6 and xy + y z + zx is equal to 12 then show that x cube + y cube + Z cube is equal to 3 x y z

Answer 1

Answer:

$\text{The value of }x^3+y^3+z^3\text{ is } 3xyz$

Step-by-step explanation:

Given x + y + z = 6 and xy + yz + zx = 12

Consider, x + y + z = 6

Squaring on both sides, we get

$(x + y + z)^2 = 36$

$x^2 + y^2 + z^2 +2(xy + yz +zx) = 36$

⇒ $x^2 + y^2 + z^2 = 36 - 2(xy + yz +zx)=36-2(12)=36-24=12$

⇒ $x^2 + y^2 + z^2 = 12$

We know that

$x^3 + y^3 + z^3 - 3xyz = (x + y + z)( x^2 + y^2 + z^2 - xy- yz - zx)$

                                    $= 6(12 -12) = 0$

⇒ $x^3 + y^3 + z^3 = 3xyz$

$\text{The value of }x^3+y^3+z^3\text{ is } 3xyz$