Question

show that; summation ab(a+b)+2abc = (a+b)(b+c)(a+c)

Answer 1

Given:

summation ab(a + b) + 2abc = (a + b) (b + c) (a + c)

To find:

show that; summation ab(a + b) + 2abc = (a + b) (b + c) (a + c)

Solution:

From given, we have,

summation ab(a + b) + 2 abc

[ ∑ ab(a + b) ] + 2 abc

= ab(a + b) + bc(b + c) + ca(c + a) + 2 abc

= a²b + ab² + b²c + bc² + c²a + ca² + 2 abc

= L.H.S

(a + b) (b + c) (a + c)

= (ab + ac + b² + bc) (a + c)

= a²b + abc + a²c + ac² + ab² + bc² + abc + bc²

= a²b + ab² + b²c + bc² + c²a + ca² + 2 abc

= R.H.S

As L.H.S = R.H.S

Hence proved.