Step-by-step explanation:
[tex] \bf \underline{Given \:Question-} \\ [/tex]
[tex]\textsf{Resolve in to factors }[/tex]
[tex] \sf \: {\bigg(\displaystyle\sum_{x,y,z}\rm x\bigg)}^{3} - \displaystyle\sum_{x,y,z}\rm {x}^{3} [/tex]
[tex] \red{\large\underline{\sf{Solution-}}}[/tex]
Given expression is
[tex]\rm :\longmapsto\: \: {\bigg(\displaystyle\sum_{x,y,z}\rm x\bigg)}^{3} - \displaystyle\sum_{x,y,z}\rm {x}^{3} [/tex]
can be rewritten as
[tex] \rm \: = \: {(x + y + z)}^{3} - ( {x}^{3} + {y}^{3} + {z}^{3})[/tex]
can be further rewritten as
[tex] \rm \: = \: {(x + y + z)}^{3} - {x}^{3} - {y}^{3} - {z}^{3}[/tex]
[tex] \rm \: = \: [{(x + y + z)}^{3} - {x}^{3}] - [{y}^{3} + {z}^{3}][/tex]
We know,
[tex]\boxed{\tt{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} + {y}^{2} - xy)}}[/tex]
and
[tex]\boxed{\tt{ {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + {y}^{2} + xy)}}[/tex]
So, using these Identities, we get
[tex] \rm =(x + y + z - x)[ {(x + y + z)}^{2} + {x}^{2} - x(x + y + z)] - (y + z)( {y}^{2} + {z}^{2} - yz)[/tex]
[tex] \rm =(y + z)[ {(x + y + z)}^{2} + {x}^{2} + {x}^{2} + xy + xz] - (y + z)( {y}^{2} + {z}^{2} - yz)[/tex]
[tex] \rm =(y + z)[ {(x + y + z)}^{2} + {2x}^{2} + xy + xz] - (y + z)( {y}^{2} + {z}^{2} - yz)[/tex]
[tex] \rm =(y + z)[ {(x + y + z)}^{2} + {2x}^{2} + xy + xz - {y}^{2} - {z}^{2} + yz][/tex]
[tex] \rm =(y + z)[ {x}^{2}+{y}^{2}+{z}^{2} + 2xy + 2yz + 2zx + {2x}^{2} + xy + xz - {y}^{2} - {z}^{2} + yz][/tex]
[tex] \rm =(y + z)[3{x}^{2} + 3xy + 3yz + 3zx][/tex]
[tex] \rm \: = \: 3(y + z)( {x}^{2} + xy + yz + zx)[/tex]
[tex] \rm \: = \: 3(y + z)[x(x + y) + z(x + y)][/tex]
[tex] \rm \: = \: 3(y + z)[(x + y)(x + z)][/tex]
[tex] \rm \: = \: 3(y + z)(x + y)(x + z)[/tex]
Hence,
[tex] \boxed{\tt{ \sf {\bigg(\displaystyle\sum_{x,y,z}\rm x\bigg)}^{3} - \displaystyle\sum_{x,y,z}\rm {x}^{3} = 3(x + y)(y + z)(z + x)}}[/tex]
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More Identities to know:-
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)² = (a - b)² + 4ab
(a - b)² = (a + b)² - 4ab
(a + b)² + (a - b)² = 2(a² + b²)
(a + b)³ = a³ + b³ + 3ab(a + b)
(a - b)³ = a³ - b³ - 3ab(a - b)
