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If alpha and beta are zeroes of the quadratic polynomial p(x)=6x²+x-1, then find the value of alpha/beta+beta/alpha+2×(1/alpha+1/beta)+3alpha×beta

Asked by mkmishra101 13th June 2016, 2:56 PM
Answered by Expert
Answer:
Since space straight alpha space and space straight beta space are space zeroes space of space the space polynomial comma space 6 straight x squared plus straight x minus 1 comma
thus comma space straight alpha plus space straight beta space equals space fraction numerator negative 1 over denominator 6 end fraction space and space αβ space equals space fraction numerator negative 1 over denominator 6 end fraction... left parenthesis straight i right parenthesis
Now comma space straight alpha over straight beta plus straight beta over straight alpha plus 2 open parentheses 1 over straight alpha plus 1 over straight beta close parentheses plus 3 αβ
equals fraction numerator straight alpha squared plus straight beta squared over denominator αβ end fraction plus 2 open parentheses fraction numerator straight alpha plus straight beta over denominator αβ end fraction close parentheses plus 3 αβ
equals fraction numerator left parenthesis straight alpha plus straight beta right parenthesis squared minus 2 αβ over denominator αβ end fraction plus 2 open parentheses fraction numerator straight alpha plus straight beta over denominator αβ end fraction close parentheses plus 3 αβ
Substitute space left parenthesis straight i right parenthesis space in space the space above space expression
fraction numerator left parenthesis straight alpha plus straight beta right parenthesis squared minus 2 αβ over denominator αβ end fraction plus 2 open parentheses fraction numerator straight alpha plus straight beta over denominator αβ end fraction close parentheses plus 3 αβ
equals fraction numerator space open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses squared minus 2 open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses over denominator open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses end fraction plus 2 open parentheses fraction numerator open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses over denominator open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses end fraction close parentheses plus 3 open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses
equals fraction numerator space open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses open square brackets fraction numerator negative 1 over denominator 6 end fraction minus 2 close square brackets over denominator open parentheses fraction numerator negative 1 over denominator 6 end fraction close parentheses end fraction plus 2 minus 1 half
equals fraction numerator negative 13 over denominator 6 end fraction plus 3 over 2 equals fraction numerator negative 4 over denominator 6 end fraction equals fraction numerator negative 2 over denominator 3 end fraction
Answered by Expert 13th June 2016, 5:03 PM
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