Answered

Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

Resolve into factor: {Σ_(x,y,z) x}³ - Σ_(x,y,z) x³.​

Resolve Into Factor Σxyz X Σxyz X class=

Sagot :

SalmonWorldTopper

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]

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

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)

Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.