Question

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`.

Answer 1

`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`