Question

Factorize each of the following expressions:
x³-8y³+27z³+18xyz

Answer 1

The algebraic expression is given as:  x³ - 8y³ + 27z³ + 18xyz

We can rewrite the  given expression as :  

= x³ + (-2y)³ + (3z)³ − 3 × x × (−2y)(3z)

In order to factorise the given algebraic expression of the form a³ + b³ + c³ - 3abc , we use the following Identity -: a³ + b³ + c³ - 3abc = (a + b + c) (a² + b³ + c² - ab - bc - ca) :  

= {(x + (−2y) + 3z)}{(x² + (−2y)² + (3z)² − x(−2y) − (−2y)(3z) −3z(x)}

= (x −2y + 3z)(x² + 4y²  + 9z² + 2xy + 6yz − 3zx)

Hence, the factorization of an algebraic expression is  (x −2y + 3z)(x² + 4y²  + 9z² + 2xy + 6yz − 3zx).

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Answer 2

$\Large{\underline{\underline{\bf{Solution :}}}}$

$\rule{200}{2}$

$\sf{\ddag \: \: \: \: \: \: Equation : \: \: x^3 - 8y^3 - 27z^3 + 18xyz \: \: \: \: \: \: }$

As, we have to factorise the Equation.

We can re - write it.

$\sf{→(x)^3 + (-2y)^3 + (3z)^3 - 3(x)(-2y)(3z)} \\ \\ \bf{We \: know \: that,} \\ \\ \footnotesize{\star{\boxed{\sf{a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca}}}}$

Where,

a is 3x

b is -y

c is -z

$\rule{150}{2}$

$\sf{→(x + (-2y) + 3z)\bigg((x)^2 + (-2y)^2 + (3z)^2 - (x)(-2y) - (-2y)(3z) - (x)(3z) \bigg)} \\ \\ \sf{→(x - 2y + 3z)(x^2 + 4y^2 + 9z^2 + 2xy + 6yz - 3xz)}$