Question

Factorize 64/125x³–8 96/25x² +48/5x​

Answer 1

Step-by-step explanation:

${\large{\rm{\underline{\red{Given \: Equation :}}}}}$

${\bold{\sf{❍ \: \: \: \: \: \Big(\frac{64}{125}\Big)x³ \: - \: 8 \: - \: \Big(\frac{96}{25}\Big)x² \: + \: \Big(\frac{48}{5}\Big)x}}}$

We will simply write in the cube format

${\bold{\sf{⟼ \: \: \: \: \: {\Big(\frac{4}{5}\Big)}^{3} \: x³ \: - \: (2)³ \: - \: 6{\Big(\frac{4}{5}\Big)}^{2} \: x² \: + \: 12\Big(\frac{4}{5}\Big)x}}}$

${\bold{\sf{⟼ \: \: \: \: \: {\Big(\frac{4x}{5}\Big)}^{3} \: - \: (2)³ \: - \: 6{\Big(\frac{4x}{5}\Big)}^{2} \: + \: 12\Big(\frac{4x}{5}\Big) \: \: ... \: Equation 1}}}$

${\bold{\rm{Let's \: assume \: \frac{4x}{5} \: = \: a}}}$

Substituting the assumed value of a in Equation 1

${\bold{\sf{⟼ \: \: \: \: \: a³ \: - \: (2)³ \: - \: 6a² \: + \: 12a}}}$

${\bold{\sf{⟼ \: \: \: \: \: a³ \: - \: (-6a²) \: + \: 12a \: - \: (2)³ \: ... \: Equation 2}}}$

The above equation is in the form of identity :

${\bold{\rm{\orange{(a + b)³ \: = \: a³ \: - \: 3a²b \: + \: 3ab² \: - \: b³}}}}$

Expressing the Equation 2 as per the identity we get,

${\bold{\sf{⟼ \: \: \: \: \: a³ \: - \: (3 × a² × 2) \: - \: (3 × a × 2²) \: + 2³ \: ... \: Equation 3}}}$

The Equation 3 is in the form of identity :

${\bold{\rm{\orange{(a - b)³ \: = \: a³ \: - \: 3ab(a - b) \: - \: b³}}}}$

Simplyfing as per the above identify we get,

${\bold{\sf{⟼ \: \: \: \: \: a³ \: - \: 6a(a - 2) \: - \: (2)³}}}$

${\bold{\sf{⟼ \: \: \: \: \: (a - 2)³}}}$

${\bold{\rm{Putting \: a \: = \: \: \frac{4x}{5} \: we \: get}}}$

${\bold{\sf{⟼ \: \: \: \: \: {\Big[\Big(\frac{4x}{5}\Big) \: - \: 2\Big]}^{3}}}}$

Taking out 2 as a common factor

${\bold{\sf{⟼ \: \: \: \: \: (2)³ \: \Big[\Big(\frac{2x}{5}\Big) \: - 1 \: \Big]³}}}$

${\bold{\sf{⟼ \: \: \: \: \: 8 {\Big(\frac{2x \: - \: 5}{5}\Big)}^{3}}}}$

${\bold{\sf{⟼ \: \: \: \: \: \frac{8}{5³} \: (2x \: - \: 5)³}}}$

Answer 2

Step-by-step explanation:

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