Question

Find the value of (x - a)³ + (x - b)³ + (x - c)³ - (x - a)(x - b)(x-c) where a + b + c = 3x

Answer 1

Step-by-step explanation:

${\Large{\mathbf{\purple{Proper \: Question \: :}}}}$

Find the value of (x - a)³ + (x - b)³ + (x - c)³ - 3(x - a)(x - b)(x-c) where a + b + c = 3x

${\Large{\mathbf{\purple{Solution \: :}}}}$

$✰ \: \: \: \: {\bold{\mathrm{\underline{We \: know \: that,}}}}$

➠ a ³ + b³ + c³ - 3abc = (a + b + c) (a² + b² + c² - ab - ac - bc)

➠ If (a + b + c) = 0 then,

➠ a³ + b³ + c³ = 3abc

Here,

$● \: \: \: \: \: \: {\bold{\sf{\orange{a \: = \: x \: - \: a}}}}$

$● \: \: \: \: \: \: {\bold{\sf{\orange{b \: = \: x \: - \: b}}}}$

$● \: \: \: \: \: \: {\bold{\sf{\orange{c \: = \: x \: - \: c}}}}$

Putting the values,

${\bold{\mathsf{⟼ \: \: \: \: \: \: a \: + \: b \: + \: c \: = \: x \: - \: a \: + \: x \: - \: b \: + \: x \: - \: c}}}$

${\bold{\mathsf{⟼ \: \: \: \: \: \: 3x \: - \: (a \: + \: b \: + \: c)}}}$

${\bold{\mathsf{⟼ \: \: \: \: \: \: 3x \: - \: 3x \: = \: 0}}}$

Thus,

(x - a)³ + (x - b)³ + (x - c)³ - 3(x - a)(x - b)(x-c) = 0

Answer 2

Answer:

Given Equation is a + b + c = 3x.

Given (x - a)^3 + (x - b)^3 + (x - c)^3 - 3(x - a)(x - b)(x - c)  

Given Equation is in the form of a^3 + b^3 + c^3 - 3abc.

We know that a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)

= > (x - a + x - b + x - c)[(x - a)^2 + (x - b)^2 + (x - c)^2 - (x - a)(x - b) - (x - b)(x - a) - (x - c)(x - a)]  

= > 3x - (a + b + c)[(x - a)^2 + (x - b)^2 + (x - c)^2 - (x - a)(x - b) - (x - b)(x - a) - (x - c)(x - a)]

= > 3x - 3x  

= > 0.

Step-by-step explanation:

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