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If a+b+c=5,and ab+bc+ca=10,prove a^3+b^3+c^3-3abc=-25

Asked by Anjali Srivastava 13th June 2016, 5:10 PM
Answered by Expert
Answer:
Recall space the space identity comma
straight a cubed plus straight b cubed plus straight c cubed minus 3 abc space equals space left parenthesis straight a space plus space straight b thin space plus space straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus ab minus bc minus ca right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight a space plus space straight b thin space plus space straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus left parenthesis ab plus bc plus ca right parenthesis right parenthesis... left parenthesis straight A right parenthesis
given space that space ab space plus space bc space plus space ca space equals space 10 space and space straight a plus straight b plus straight c space equals space 5
Now comma space left parenthesis straight a space plus space straight b thin space plus space straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus left parenthesis ab plus bc plus ca right parenthesis right parenthesis equals 5 left parenthesis straight a squared plus straight b squared plus straight c squared minus 10 right parenthesis

left parenthesis straight a plus straight b plus straight c right parenthesis squared equals straight a squared plus straight b squared plus straight c squared plus 2 ab plus 2 bc plus 2 ca
rightwards double arrow 5 squared equals straight a squared plus straight b squared plus straight c squared plus 2 left parenthesis ab plus bc plus ca right parenthesis
rightwards double arrow 25 equals straight a squared plus straight b squared plus straight c squared plus 2 left parenthesis 10 right parenthesis
rightwards double arrow 25 equals straight a squared plus straight b squared plus straight c squared plus 20
rightwards double arrow straight a squared plus straight b squared plus straight c squared equals 5... left parenthesis straight i right parenthesis
So comma space substitute space left parenthesis straight i right parenthesis space in space left parenthesis straight A right parenthesis comma space we space get
space left parenthesis straight a space plus space straight b thin space plus space straight c right parenthesis left parenthesis straight a squared plus straight b squared plus straight c squared minus left parenthesis ab plus bc plus ca right parenthesis right parenthesis equals 5 left parenthesis 5 minus 10 right parenthesis equals 5 left parenthesis negative 5 right parenthesis equals negative 25
Answered by Expert 13th June 2016, 6:24 PM
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