Question

a³-8b³-64c³-24abc factorise the following
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Answer 1

$\mathfrak{\large{\underline{\underline{Answer:-}}}}$

$\boxed{\bf{(a -2b - 4c)( {a}^{2} + 4 {b}^{2} + 16 {c}^{2} + 2ab - 8bc + 4ac)}}$

$\mathfrak{\large{\underline{\underline{Explantion:-}}}}$

${a}^{3} - 8 {b}^{3} - 64 {c}^{3} - 24abc$

It can be written as :-

$= {a}^{3} + {(2b)}^{3} + {( - 4c)}^{3} - 3(a)( - 2b)( - 4c)$

This expression can be factorised by using an alebraic identity i.e, x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)

Here x = a , y = - 2c, z = - 4c

By substituting the values we have :-

$= (a + ( - 2b) + ( - 4c))( {a}^{2} + {( - 2c)}^{2} + {( - 4c)}^{2} - (a)( - 2b) - ( - 2b)( - 4c) - (a)( - 4c))$

$= (a -2b - 4c)( {a}^{2} + 4 {b}^{2} + 16 {c}^{2} + 2ab - 8bc + 4ac)$

$\boxed{\bf{(a -2b - 4c)( {a}^{2} + 4 {b}^{2} + 16 {c}^{2} + 2ab - 8bc + 4ac)}}$

$\mathfrak{\large{\underline{\underline{Identity\:Used:-}}}}$

x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)

$\mathfrak{\large{\underline{\underline{Extra\:Information:-}}}}$

What is factorisation ?

Factorization is a process of writing the given expression as product of its factors.

1) Factorization by grouping terms :-

Example :- ax + bx + ay + byat

= x(a + b) + y(a + b)

= (a + b)(x + y)

2) Factorisation using identities

Example : 25p² - 49q²

= (5p)² - (7q)²

= (5p + 7q)(5p - 7q) [Since x² + y² = (x + y)(x - y)

What is identity ?

An equation is called an identity if it is satisfied by any value that replaces its variables.