So Simple,,,,,.....

Algebra Level 2

If a + b + c = 9 a+b+c = 9 and a b + b c + c a = 26 ab+bc+ca = 26 , then what is the value of

a 3 + b 3 + c 3 − 3 a b c ? a^3+b^3+c^3 - 3abc ?

29 729 27 495

Tushar Malik by Tushar Malik , 20, India

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1 solution

Tom Engelsman
Sep 1, 2017

Let's cube the first equation out and simplify from there:

( a + b + c ) 3 = 9 3 ; (a+b+c)^3 = 9^3;

or ( a + b ) 3 + 3 ( a + b ) 2 c + 3 ( a + b ) c 2 + c 3 = 729 ; (a+b)^3 + 3(a+b)^{2}c + 3(a+b)c^2 + c^3 = 729;

or ( a 3 + 3 a 2 b + 3 a b 2 + b 3 ) + 3 ( a 2 + 2 a b + b 2 ) c + 3 ( a + b ) c 2 + c 3 = 729 ; (a^3 + 3a^{2}b + 3ab^2 + b^3) + 3(a^2 + 2ab + b^2)c + 3(a+b)c^2 + c^3 = 729;

or ( a 3 + b 3 + c 3 ) + 3 a b ( a + b ) + 3 a c ( a + c ) + 3 b c ( b + c ) + 6 a b c = 729 ; (a^3 + b^3 + c^3) + 3ab(a+b) + 3ac(a+c) + 3bc(b+c) + 6abc = 729;

or ( a 3 + b 3 + c 3 ) + 3 a b ( 9 − c ) + 3 a c ( 9 − b ) + 3 b c ( 9 − a ) + 6 a b c = 729 ; (a^3 + b^3 + c^3) + 3ab(9-c) + 3ac(9-b) + 3bc(9-a) + 6abc = 729;

or ( a 3 + b 3 + c 3 ) + 27 ( a b + a c + b c ) − 9 a b c + 6 a b c = 729 ; (a^3 + b^3 + c^3) + 27(ab + ac + bc) - 9abc + 6abc = 729;

or ( a 3 + b 3 + c 3 ) + 27 ( 26 ) − 3 a b c = 729 ; (a^3 + b^3 + c^3) + 27(26) - 3abc = 729;

or ( a 3 + b 3 + c 3 ) − 3 a b c = 729 − ( 26 ) ( 27 ) = 27 . (a^3 + b^3 + c^3) - 3abc = 729 - (26)(27) = \boxed{27}.

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