IF A+B+C=5  AND AB+BC+CA=10 THEN PROVE THAT A3+B3+C3 -3ABC =-25

Asked by vrishti TOKAS | 20th May, 2015, 08:05: PM

Expert Answer:

G i v e n space t h a t space A plus B plus C equals 5 space a n d space A B plus B C plus C A equals 10 W e space n e e d space t o space p r o v e space t h a t space A cubed plus B cubed plus C cubed minus 3 A B C equals negative 25 C o n s i d e r space t h e space i d e n t i t y A cubed plus B cubed plus C cubed minus 3 A B C equals open parentheses A plus B plus C close parentheses open parentheses A squared plus B squared plus C squared minus A B minus B C minus C A close parentheses T h u s comma space s u b s t i t u t i n g space t h e space v a l u e s space A plus B plus C equals 5 space a n d space A B plus B C plus C A equals 1 comma space w e space h a v e comma A cubed plus B cubed plus C cubed minus 3 A B C equals 5 cross times open square brackets A squared plus B squared plus C squared minus A B minus B C minus C A close square brackets rightwards double arrow A cubed plus B cubed plus C cubed minus 3 A B C equals 5 cross times open square brackets open parentheses A squared plus B squared plus C squared close parentheses minus open parentheses A B plus B C plus C A close parentheses close square brackets rightwards double arrow A cubed plus B cubed plus C cubed minus 3 A B C equals 5 cross times open square brackets open parentheses A squared plus B squared plus C squared close parentheses minus 10 close square brackets... left parenthesis 1 right parenthesis W e space n e e d space t o space f i n d space t h e space v a l u e space o f space A squared plus B squared plus C squared colon A squared plus B squared plus C squared equals open parentheses A plus B plus C close parentheses squared minus 2 open parentheses A B plus B C plus C A close parentheses rightwards double arrow A squared plus B squared plus C squared equals 5 squared minus 2 cross times 10 rightwards double arrow A squared plus B squared plus C squared equals 25 minus 20 rightwards double arrow A squared plus B squared plus C squared equals 5 T h e r e f o r e comma space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma A cubed plus B cubed plus C cubed minus 3 A B C equals 5 cross times open square brackets 5 minus 10 close square brackets rightwards double arrow A cubed plus B cubed plus C cubed minus 3 A B C equals 5 cross times open square brackets negative 5 close square brackets rightwards double arrow A cubed plus B cubed plus C cubed minus 3 A B C equals negative 25

Answered by Vimala Ramamurthy | 21st May, 2015, 11:04: AM

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