If alpha, beta and gama are the zeros of the polynomial ax3 + bx2 + cx + d, find the value of 1 / alpha 2 + 1 / beta 2 + 1 / gama2

Asked by deepa1973 | 14th Jun, 2015, 09:49: AM

Expert Answer:

a x cubed plus b x squared plus c x plus d  S i n c e space alpha comma beta comma gamma space a r e space t h e space r o o t s space o f space t h e space e q u a t i o n comma alpha plus beta plus gamma equals negative b over a alpha beta plus beta gamma plus alpha gamma equals c over a alpha beta gamma equals negative d over a 1 over alpha squared plus 1 over beta squared plus 1 over gamma squared equals fraction numerator alpha squared beta squared plus beta squared gamma squared plus alpha squared gamma squared over denominator alpha squared beta squared gamma squared end fraction equals fraction numerator open parentheses alpha beta plus beta gamma plus alpha gamma close parentheses squared minus 2 open parentheses alpha beta squared gamma plus alpha beta gamma squared plus alpha squared beta gamma close parentheses over denominator alpha squared beta squared gamma squared end fraction space left square bracket U sin g space left parenthesis a plus b plus c right parenthesis squared equals a squared plus b squared plus c squared plus 2 left parenthesis a b plus b c plus c a 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 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 space space space space space space space space rightwards double arrow a squared plus b squared plus c squared equals left parenthesis a plus b plus c right parenthesis squared minus 2 left parenthesis a b plus b c plus c a right parenthesis right square bracket equals fraction numerator open parentheses alpha beta plus beta gamma plus alpha gamma close parentheses squared minus 2 alpha beta gamma open parentheses alpha plus beta plus gamma close parentheses over denominator open parentheses alpha beta gamma close parentheses squared end fraction equals fraction numerator open parentheses c over a close parentheses squared minus 2 open parentheses negative d over a close parentheses open parentheses negative b over a close parentheses over denominator open parentheses negative d over a close parentheses squared end fraction equals fraction numerator begin display style c squared over a squared minus fraction numerator 2 b d over denominator a squared end fraction end style over denominator begin display style d squared over a squared end style end fraction equals fraction numerator begin display style c squared minus 2 b d end style over denominator begin display style d squared end style end fraction

Answered by Prasenjit Paul | 15th Jun, 2015, 10:48: AM

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