Simplify:
(a² - b²)³ + (b² - c²)³ + (c² - a²)³ / (a - b)³ + (b - c)³ + (c - a)³.
Show answer & explanation
Correct answer: A
Explanation
Given:
((a² − b²)³ + (b² − c²)³ + (c² − a²)³) / ((a − b)³ + (b − c)³ + (c − a)³)
By formula:
If x + y + z = 0
then x³ + y³ + z³ = 3xyz
x = a − b
y = b − c
z = c − a
(a − b) + (b − c) + (c − a) = 0
(a − b)³ + (b − c)³ + (c − a)³ = 3(a − b)(b − c)(c − a)
Now in the numerator
a² − b² = (a − b)(a + b)
b² − c² = (b − c)(b + c)
c² − a² = (c − a)(c + a)
(a² − b²) + (b² − c²) + (c² − a²) = 0
(a² − b²²³ + (b² − c²²³) + (c² − a²²³) = 3(a² − b²)(b² − c²)(c² − a²)
= 3(a − b)(a + b)(b − c)(b + c)(c − a)(c + a)/ 3(a − b)(b − c)(c − a)
= (a + b)(b + c)(c + a)