Question

verify that x3+Y3+Z3=1/2(x-y+z)[(x-y)2whole square +(Y-Z)2WHOLE SQUARE+(Z-X)2 WHOLE SQUARE]
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Answer 1

Answer and Step-by-step explanation:

Given

LHS = x³ + y³ + z³ - 3xyz

RHS = $\frac{1}{2} (x-y-z)[(x-y)^{2} +(y-z)^{2} +(z-x)^{2} ]$

To prove

LHS = RHS

Proof

RHS

$\Rightarrow {\frac{1}{2}(x-y-z)[(x-y)^{2} +(y-z)^{2} + (z-x)^{2}] }$

$\Rightarrow\frac{1}{2}(x-y-z) [(x^2 + y^2 -2xy)+(y^2+z^2-2yz)+(z^2+x^2-2xz)]$

$\Rightarrow\frac{1}{2}(x-y-z) (2x^2+2y^2 +2z^2-2xy-2yz-2xz)$

$\Rightarrow\frac{1}{2}(x-y-z) \times2(x^2+y^2+z^2-xy-yz-xz)$

$\Rightarrow\frac{2}{2}(x-y-z) (x^{2} +y^2+z^2-xy-yz-xz)$

$\Rightarrow(x-y-z) (x^2+y^2+z^2-xy-yz-xz)$

$\Rightarrow x(x^2+y^2+z^2-xy-yz-xz)-y(x^2+y^2+z^2-xy-yz-xz)-z(x^2+y^2+z^2-xy-yz-xz)$

$\Rightarrow x^3+xy^2+xz^2-x^2y-xyz-x^2z-x^2y-y^3-yz^2+xy^2+y^2z+xyz-x^2z-y^2z-z^3+xyz+yz^2+xz^2$

$\Rightarrow x^3+y^3+z^3-3xyz$  (All the other terms get subtracted or cancelled off)

Answer 2

$\large \rm\red {\underline {\underline{To\:Prove:-}}}$

• $\rm {x}^{3} + {y}^{3} + {z}^{3} -3xyz = \dfrac{1}{2} (x - y + z) \big \{ {(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2} \big \}$

$\large \rm \green {\underline {\underline{Proof:-}}}$

$\rightarrow \rm {x}^{3} + {y}^{3} + {z}^{3} -3xyz = \dfrac{1}{2} (x + y + z) \big \{ {(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2} \big \}$

$\rm Let \: us \: verify \: by \: simplifying \: LHS \: and \: RHS \: separately$

RHS =

$\rightarrow \rm \dfrac{1}{2} (x + y + z) \big \{ {(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2} \big \}$

$\rightarrow \rm \dfrac{1}{2} (x + y + z) \big \{ {( {x}^{2} + {y}^{2} - 2xy )} + {(y }^{2} + {z}^{2} - 2yz ) + {( {z}^{2} + {x}^{2} - 2xz )} \big \}$

$\rightarrow \rm \dfrac{1}{2} (x + y + z) \big \{ { {x}^{2} + {y}^{2} - 2xy } + {y }^{2} + {z}^{2} - 2yz + { {z}^{2} + {x}^{2} - 2xz } \big \}$

$\rightarrow \rm \dfrac{1}{2} (x + y + z) \big \{ {2( {x}^{2} + {y}^{2} + {z}^{2} - xy } - yz - xz ) \big \}$

$\rightarrow \rm \dfrac{1}{2} (x + y + z) \big \{ {{x}^{2} + {y}^{2} + {z}^{2} - xy } - yz - xz \big \}$

$\rightarrow \rm (x + y + z) \big \{ {{x}^{2} + {y}^{2} + {z}^{2} - xy } - yz - xz \big \}$

$\rightarrow \rm x({{x}^{2} + {y}^{2} + {z}^{2} - xy } - yz - xz) + y({{x}^{2} + {y}^{2} + {z}^{2} - xy } - yz - xz) + z({{x}^{2} + {y}^{2} + {z}^{2} - xy } - yz - xz)$

$\rightarrow \rm {{x}^{3} + x{y}^{2} + x{z}^{2} - {x}^{2} y } - xyz - {x}^{2} z + {{x}^{2}y + {y}^{3} + y{z}^{2} - x {y}^{2} } - {y}^{2} z - xyz + {{x}^{2}z + {y}^{2}z + {z}^{3} - xyz } - y {z}^{2} - x {z}^{2}$

$\rightarrow \rm {x}^{3} + {y}^{3} + {z}^{3} - 3xyz = LHS$

LHS = RHS

$\large \sf {\underline{ \underline\purple{Hence \: Proved}}}$