Question

(a - b)^3+ (b-c)^3 + (c-a)^3 = 3 (a - b) (b-c) (c-a)​

Answer 1

Answer:

Step-by-step explanation:

(a-b)3 + (b-c)3 + (c-a)3 = a3 - b3 - 3ab(a-b) + b3 - c3 - 3bc(b-c) + c3 - a3 - 3ca(c-a)

= - 3a2b + 3ab2 - 3b2c + 3bc2 - 3ac2 + 3 a2c = 3 (- a2b + ab2 - b2c + bc2 - ac2 + a2c)

= 3 [(a2(c-b) + (b2(a-c) + (c2(b-a)]

Or

Let x = (a – b), y = (b – c) and z = (c – a)

Consider, x + y + z = (a – b) + (b – c) + (c – a) = 0

⇒ x3 + y3 + z3 = 3xyz

That is (a – b)3 + (b – c)3 + (c – a)3 = 3(a – b)(b – c)(c – a)

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Answer 2

Step-by-step explanation:

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

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Expert Answers info

LIX LEMJAY eNotes educator | CERTIFIED EDUCATOR

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 - 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

Answer: 0