Question

factorise 8 a cube minus b cube + 64 C + 24ab. c

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Answer 1

Answer:

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Answer 2

$\Large{\boxed{\underline{\overline{\mathfrak{\star \: Question:- \: \star}}}}}$

Factorise 8 a cube minus b cube + 64 C + 24 ABC.

$\huge\underline{\overline{\mid{\bold{\pink{ANSWER-}}\mid}}}$

We have to factorise:

$8a^3-b^3+64c^3+24abc$

For it, we will use the following identity:

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

We can factorize using the identity as follows:

$8a^3-b^3+64c^3+24abc \\ \\ = (2a)^3 + (-b)^3 + (4c)^3 -3(2a)(-b)(4c) \\ \\ = (2a-b+4c)((2a)^2+(-b)^2+(4c)^2-(2a)(-b)-(-b)(4c)-(4c)(2a)) \\ \\ = (2a-b+4c)(4a^2+b^2+16c^2+2ab+4bc-8ca)$

Thus, The factorized form is

$\bold{(2a-b+4c)(4a^2+b^2+16c^2+2ab+4bc-8ca)}$

✾ Extra Related Information :-

➹ What are the algebraic identities?

An algebraic identity is an equality that holds for any values of its variables.

For example,

The identity ( x + y )² = x ² + 2 x y + y ²

holds for all values of x and y.

➹ What are the Standard Algebraic Identities?

☆ Standard Algebraic Identities List ☆

$\sf\underline{\purple{\:\:\: \: \: Identity\: 1 :- \:\:\:}}$ ( a + b )² = a²+ 2ab + b²

$\sf\underline{\pink{\:\:\: \: \: Identity\: 2 :- \:\:\:}}$ ( a – b )² = a² – 2ab + b²

$\sf\underline{\purple{\:\:\: \: \: Identity\: 3 :- \:\:\:}}$ a² – b² = (a + b)(a – b)

$\sf\underline{\pink{\:\:\: \: \: Identity\: 4 :- \:\:\:}}$ ( x + a )( x + b ) = x² + ( a + b ) x + ab.

$\sf\underline{\purple{\:\:\: \: \: Identity\: 5 :-\:\:\:}}$( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca.

$\sf\underline{\pink{\:\:\: \: \: Identity\: 6 :- \:\:\:}}$ ( a + b )³ = a³ + bb³ + 3ab (a + b)

$\sf\underline{\purple{\:\:\: \: \: Identity\: 7 :- \:\:\:}}$ ( a – b )³ = a³ – b³– 3ab ( a –b)

$\sf\underline{\pink{\:\:\: \: \: Identity\: 8 :- \:\:\:}}$ a³ + b³ + c³ – 3abc = ( a + b + c )( a² + b² + c² – ab – bc – ca )