Question

if a, b and c are positive integers, then (a-b-c)^3-a^3+b^3+c^3 is always divisible by​

Answer 1

Answer:

option 3 is correct

3(a-b) is a factor

Step-by-step explanation:

Using the identity

(x+y+z)³= x³+y³+z³+3(x+y)(y+z)(z+x)

putting x = a  , y = - b  z = - c

=> (a - b - c)³ = a³ + (-b)³ +(-c)³  + 3(a + (-b))(-b + (-c))(-c + a)

=>  (a - b - c)³ = a³ -b³ -c³  + 3(a - b)(-b -c)(-c + a)

=>  (a - b - c)³ = a³ -b³ -c³  - 3(a - b)(b + c)(a - c))

=>  (a - b - c)³  - a³ + b³ + c³ = - 3(a - b)(b + c)(a - c)

a + b is not the factor

b + c is not the only factor

3(a-b) is a factor

3(a+b)(b+c)(a+c) is not the factor

Hence option 3  :  3(a-b) is a factor