# how to simplify:(a-b)³ + (b-c)³ + (c-a)³ -3(a-b)(b-c)(c-a)?

To simplify the above expressions, start by expanding the binomials.

Note that we  can expand the (a-b)^3 , (b-c)^3 , and (c-a)^3 using the special product formulas for a cube of a binomial.

The formula is:  (x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3

• Expand: (a-b)^3  = a^3 - 3a^2b + 3ab^2 - b^3
• Expand: (b-c)^3  = b^3 - 3b^2c + 3bc^2 - c^3
• Expand: (c-a)^3  = c^3 -  3c^2a + 3ca^2 - a^3

= c^3 - 3ac^2  + 3a^2c - a^3

Also, expand -3(a-b)(b-c)(c-a) using distributive property.

• Multiply (a-b) and (b-c).

(a-b)(b-c) = ab - ac - b^2 - b(-c)

= ab - ac - b^2 + bc

• Then, multiply (ab - ac - b^2 + bc) and (c-a).

(c - a)(ab - ac - b^2 + bc) = abc - ac^2 - b^2c + bc^2 -a^2b

+ a^2c + ab^2 - abc

= -ac^2 - b^2c + bc^2 - a^2b +

a^2c + ab^2

• Then, multiply -3 and (-ac^2 - b^2c + bc^2 - a^2b + a^2c +  ab^2).

-3(-ac^2 - b^2c + bc^2 - a^2b + a^2c + ab^2)

= 3ac^2 + 3b^2c - 3bc^2 + 3a^2b - 3a^2c - 3ab^2

Next, combine like terms of the above expanded expressions.

a^3 - 3a^2b + 3ab^2 - b^3

+ b^3 - 3b^2c + 3bc^2 - c^3

-a^3                                                           +c^3 -3ac^2 +3a^2c

+ 3a^2b - 3ab^2          + 3b^2c - 3bc^2         +3ac^2 -3a^2c

---------------------------------------------------------------------

0   +  0       + 0       + 0     + 0        + 0        + 0   + 0      + 0

Approved by eNotes Editorial

You should remember how to expand the cube of binomial such that:

`(a-b)^3 = a^3 - 3a^2*b + 3ab^2 - b^3`

`(a-b)^3 = a^3- b^3 - 3ab(a-b)`

`(b-c)^3 = b^3 - c^3 - 3bc(b-c) `

`(c-a)^3 = c^3 - a^3 - 3ac(c-a) `

`(a-b)^3 + (b-c)^3 + (c-a)^3 = a^3 - b^3 + b^3 - c^3 +c^3 - a^3 - 3(ab(a-b) + bc(b-c) + ac(c-a))`

Reducing like terms yields:

`(a-b)^3 + (b-c)^3 + (c-a)^3 = - 3(ab(a-b) + bc(b-c) + ac(c-a))`

Subtracting 3(a-b)(b-c)(c-a) yields:

`(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = - 3(ab(a-b) + bc(b-c) + ac(c-a)) - 3(a-b)(b-c)(c-a)`

Factoring out -3 yields:

`(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = - 3(ab(a-b) + bc(b-c) + ac(c-a) + (a-b)(b-c)(c-a))`

Opening the brackets to the right side yields:

`(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = -3(a^2b - ab^2 + b^2c - bc^2 + ac^2 - a^2c + (ab-ac-b^2+bc)(c-a))`

`(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = -3(a^2b - ab^2 + b^2c - bc^2 + ac^2 - a^2c + abc - a^2b - ac^2 + a^2c - b^2c + ab^2 + bc^2 - abc)`

Reducing like terms yields:

`(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = -3*0`

`(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = 0`

Hence, simplifying the given expression yields: `(a-b)^3 + (b-c)^3 + (c-a)^3 - 3(a-b)(b-c)(c-a) = 0.`