Manipulate the Binomials

Algebra Level 3

Let ( a − b ) ( b − c ) ( c − a ) = ( a + b ) ( b + c ) ( c + a ) (a-b)(b-c)(c-a)=(a+b)(b+c)(c+a)

Determine a a + b + b b + c + c c + a \frac { a }{ a+b } +\frac { b }{ b+c } +\frac { c }{ c+a }


The answer is 1.

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Watcharapon Wisadekul by Watcharapon Wisadekul , 28, Thailand

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1 solution

L e t x = a 2 b + b 2 c + c 2 a , y = a b 2 + b c 2 + c a 2 a n d z = a b c T h e f i r s t c o n d i t i o n i s x + z = 0 T h e s e c o n d e x p r e s s i o n i s 2 x + y + 3 z x + y + 2 z = 1 + x + z x + y + 2 z = 1 : ) ) Let\quad x=a^{ 2 }b+b^{ 2 }c+c^{ 2 }a,\quad y=ab^{ 2 }+bc^{ 2 }+ca^{ 2 }\quad and\quad z=abc\quad \\ The\quad first \quad condition\quad is\quad x+z=0\quad \\ The\quad second \quad expression\quad is\quad \frac { 2x+y+3z }{ x+y+2z } =1+\frac { x+z }{ x+y+2z } =1 :))

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L e t x = a 2 b + b 2 c + c 2 a , y = a b 2 + b c 2 + c a 2 a n d z = a b c T h e f i r s t c o n d i t i o n i s x + z = 0 T h e s e c o n d e x p r e s s i o n i s 2 x + y + 3 z x + y + 2 z = 1 + x + z x + y + 2 z = 1 : ) ) Let\quad x=a^{ 2 }b+b^{ 2 }c+c^{ 2 }a,\quad y=ab^{ 2 }+bc^{ 2 }\\+ca^{ 2 }\quad and\quad z=abc\quad \ The\quad first\ condition\quad\\ is\quad x+z=0\quad \ The\quad second\ expression\quad\\ is\quad \frac { 2x+y+3z }{ x+y+2z } =1+\frac { x+z }{ x+y+2z } =1 :))

Adarsh Kumar - 6 years, 7 months ago

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thx my fri :)))

Watcharapon Wisadekul - 6 years, 7 months ago

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put \ ( and \ ) (without spaces) before and after the LaTeX code

Happy Melodies - 6 years, 6 months ago

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