Question

Verify that x^3+y^3+z^3-3xyz=1/2 (x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]

Plz answer

Answer 1

$\large\bf\underline \pink{Question:-}$

x³ + y³ + z³ - 3xyz = ½ (x+y+z) [(x-y)² +. (y-z)² + (z-x)²]

$\huge\rm\underline \red{Verification:-}$

$\\ \large\bf \red {\dag \: {\underline{ \blue{ RHS : \: \: \: }}}}$

$\mapsto\rm\: \frac{1}{2}(x + y + z)[(x - y {)}^{2} + (y - z {)}^{2} + (z - x {)}^{2} ] \\ \\ \small\mapsto\rm\: \frac{1}{2}(x + y + z)[( {x}^{2} + {y}^{2} - 2xy ) + ( {y}^{2} + {z}^{2} - 2yz ) +( {z}^{2} + {x}^{2} - 2xz ) ] \\ \\ \mapsto\rm\small \frac{1}{2}(x + y + z) [ {x}^{2} + {x}^{2} + {y}^{2} + {y}^{2} + {z}^{2} + {z}^{2} - 2xy - 2yz - 2xz] \\ \\ \mapsto\rm\small \frac{1}{2} (x + y + z)[2 {x}^{2} + 2 {y}^{2} + 2 {z}^{2} - 2xy - 2yz - 2xz ] \\ \\\mapsto\rm\small \frac{1}{ \cancel2}(x + y + z) \times \cancel 2 [ {x}^{2} + {y}^{2} + {z}^{2} - xy - yz - xz ] \\ \\ \mapsto\rm\:(x + y + z) [{x}^{2} + {y}^{2} + {z}^{2} - xy - yz - xz ] \\ \\ \mapsto\rm\small\:x({x}^{2} + {y}^{2} + {z}^{2} - xy - yz - xz) + y({x}^{2} + {y}^{2} + {z}^{2} - xy - yz - xz) \\ \rm \: \: \: \: \: \: + z({x}^{2} + {y}^{2} + {z}^{2} - xy - yz - xz) \\ \\ \mapsto \rm {x}^{3} +x y {}^{2} +x z {}^{2} - {x}^{2}y - xyz - {x}^{2} z \\ \: \rm \small + {x}^{2}y + {y}^{3} + yz {}^{2} - x {y}^{2} - {y}^{2}z - xyz \\ \: \rm \small+ {x}^{2} z + {y}^{2}z + {z}^{3} - xyz - y {z}^{2} - x {z}^{2} \\ \\ \mapsto\rm\small \: {x}^{ 3} + {y}^{3} + {z}^{3} - 3xyz + xy {}^{2} - x {y}^{2} + x {z}^{2} \\ \rm \small \: \: - x {z}^{2} - {x}^{2}y + {x}^{2}y - {x}^{2} z + {x}^{2}z + yz {}^{2} - yz {}^{2} \\ \\ \mapsto\bf\: {x}^{ 3} + {y}^{3} + {z}^{3} - 3xyz$

LHS = RHS

Hence verified .

Answer 2

✯✯ QUESTION ✯✯

Verify that $\tt{x}^{3}+{y}^{3}+{z}^{3}-3xyz=\dfrac{1}{2}(x+y+z)[{(x-y)}^{2}+{(y-z)}^{2}+{(z-x)}^{2}]$

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✰✰ ANSWER ✰✰

➥R.H.S : -

$\longrightarrow\tt{\dfrac{1}{2}(x+y+z)[{(x-y)}^{2}+{(y-x)}^{2}+{(z-x)}^{2}]}$

$\longrightarrow\tt{\dfrac{1}{2}(x+y+z)[{x}^{2}+{y}^{2}-2xy+{y}^{2}+{z}^{2}-2yz+{z}^{2}+{x}^{2}-2xz]}$

$\longrightarrow\tt{\dfrac{1}{2}(x+y+z)[{2x}^{2}+{2y}^{2}+{2z}^{2}-2xy-2yz-2xz]}$

$\longrightarrow\tt{\dfrac{1}{\cancel{2}}(x+y+z)\cancel{(2)}{(x}^{2}+{y}^{2}+{z}^{2}-xy-yz-xz)}$

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

$\longrightarrow\tt{{x}^{3}+{y}^{3}+{z}^{3}-3xyz}$

$\red\longmapsto\:\large\underline{\boxed{\bf\green{L.H.S}\orange{=}\purple{R.H.S}}}$

✺ HENCE VERIFIED ✺