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If `alpha` and `beta` are the zeros of the quadratic polynomial `p(t)=t^2-5t-1` , find the value of `alpha^2/beta^2 + beta^2/alpha^2 +2(alpha/beta+beta/alpha)-alphabeta`. |
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Answer» `alpha and beta` are the roots of polynomial `t^2-5t-1`. `:. alpha+beta = -(-5)/1 = 5` `alpha beta = -1/1 = -1` Now,`alpha^2/beta^2+beta^2/alpha^2+2(alpha/beta+beta/alpha)-alphabeta = (alpha^4+beta^4)/(alpha^2beta^2)+2(alpha^2+beta^2)/(alphabeta)-alphabeta` `=((alpha^2+beta^2)^2-2alpha^2beta^2)/(alphabeta)^2+(2((alpha+beta)^2-2alphabeta))/(alphabeta)-alphabeta` `=(((alpha+beta)^2-2alphabeta)^2-2(alphabeta)^2)/(alphabeta)^2+(2((alpha+beta)^2-2alphabeta))/(alphabeta)-alphabeta` `=((25+2)^2-2(1))/1 +(2(25+2))/(-1)+1` `=727-54+1 = 674` |
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