If `alpha` and `beta` are zeroes of the quadratic polynomial f(x)`=3x^(2)-5x-2`, then find the value of `((alpha^(2))/(beta)+(beta^(2))/(alpha))+6(alpha+1)(beta+1)`.

Answer

N/a

Explanation

Since `alpha` and `beta` are zeroes of `3x^(2)-5x-2`
`:. " " alpha+beta=-(b)/(a)=(-(-5))/(3)=(5)/(3),alphabeta=(c )/(a)=(-2)/(3)`
`:. " "(alpha^(2))/(beta)+(beta^(2))/(alpha)=(alpha^(3)+beta^(3))/(alphabeta)=((alpha+beta)^(3)-3alphabeta(alpha+beta))/(alphabeta)`
`=(((5)/(3))^(3)-3(-(2)/(3))((5)/(3)))/(-(2)/(3))=((125)/(27)+(10)/(3))/(-(2)/(3))`
`=((125+90)/(27))(-(3)/(2))=-(215)/(18) " "`......(1)
and `" " (alpha+1)(beta+1)=alphabeta+alpha+beta+1=-(2)/(3)+(5)/(3)+1=2`
`:. " " 6(alpha+1)(beta+1)=12 " "`.....(2)
`:. ((alpha^(2))/(beta)+(beta^(2))/(alpha))+6(alpha+1)(beta+1)=(-215)/(18)+12=(1)/(18) " "` Ans.