Simplify the following expression (x - y)³ + (y - z)³ + (z -x)³ over (x - y)(y - z)(z - x) .

Simplify the following expression (x - y)³ + (y - z)³ + (z…

Simplify the following expression

${{(x - y)^3 + (y - z)^3 + (z -x)^3} \over (x - y)(y - z)(z - x)}$ .

Show answer & explanation

Correct answer: A

Explanation

The correct answer is option 1 i.e. 3

${{(x - y)^3 + (y - z)^3 + (z -x)^3} \over (x - y)(y - z)(z - x)}$

The numerator contains the sum of cubes:

a³+ b³+ c³ = (a + b + c)(a² + b² + c² − ab − bc − ca)

where a = (x − y), b = (y − z), and c = (z − x)

a + b + c = (x − y) + (y − z) + (z − x) = 0

the sum of cubes = a³ + b³ + c³ = 3abc

Hence

(3⋅(x − y)(y − z)(z − x))/((x - y) (y - z) (z - x)) = 3