Question

Show that ab(a  b)  2abc  (a  b)(b  c)(c  a)

Answer 1

Given:

In the question, there is a mistyping error. so, correct question is:

∑ab(a+b)+2abc = (a+b)(b+c)(a+c)

To find:

prove.

Solution:

∑ab(a+b)+2abc = (a+b)(b+c)(a+c)

Formula:

$\to \bold{ \sum ab(a+b) =ab(a + b) + bc(b + c) + ca(c + a)}$

Solve the L.H.S part:

$\to ab(a + b) + bc(b + c) + ca(c + a)+2abc\\\\\to \bold{a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2+2abc}$

Solve the R.H.S part:

$\to (a+b)(b+c)(a+c)\\\\\to (ab+ac+b^2+bc)(a+c)\\\\\to a^2b+a^2c+ab^2+abc +abc+ac^2+b^2c+bc^2\\\\ \to \bold{a^2b + ab^2 + b^2c + bc^2 + c^2a + ca^2+2abc}$

The final answer is "L.H.S = R.H.S".