If `alpha,beta` are roots of `x^2-px+q=0` then find the quadratic equation whose roots are `((alpha^2-beta^2)(alpha^3-beta^3))` and `alpha^2beta^3+alpha^3beta^2`

Explanation

Since `alpha, beta` are the roots of `x^(2)-px+q=0`
`:. alpha+beta=p,alphabeta=q`
`impliesalpha-beta=sqrt((p^(2)-4q))`
Now `(alpha^(2)-beta^(2))(alpha^(3)-beta^(3))`
`=((alpha+beta)(alpha-beta)(alpha-beta)(alpha^(2)+alpha beta+ beta^(2))`
`=(alpha+beta)(alpha-beta)^(2){(alpha+beta)^(2)-alpha beta}`
`=p(p^(2)-4q)(p^(2)-q)`
and `alpha^(3)beta^(2)+alpha^(2)beta^(3)=alpha^(2)beta^(2)(alpha+beta)=pq^(2)`
`S=` Sum of roots `=p(p^(2)-4q)(p^(2)-q)+pq^(2)`
`=p(p^(4)-5p^(2)q+5q^(2))`
`P=` Product of roots `=p^(2)q^(2)(p^(2)-4q)(p^(2)-q)`
`:.` Required equation is `x^(2)-Sx+P=0`
i.e. `x^(2)-p(p^(4)-5p^(2)q+5q^(2))x+p^(2)q^(2)(p^(2)-4q)(p^(2)-q)=0`