If `alpha` and `beta` are zeroes of the quadratic polynomial `2x^(2)+2(a+b)x+a^(2)+b^(2)`, form the quadratic polynomial whose zeroes are `(alpha+beta)^(2)`.

Answer

N/a

Explanation

Since `alpha` and `beta` are zeroes
`:. " " alpha+beta=-(2(a+b))/(2)=-(a+b),alphabeta=(a^(2)+b^(2))/(2)`
Polynomials
Now,`" " S=(alpha+beta)^(2)+(alpha-beta)^(2)`
`=alpha^(2)+beta^(2)+2alphabeta+alpha^(2)+beta^(2)-2alphabeta`
`=2(alpha^(2)+beta^(2))=2[(alpha+beta)^(2)-2alpha beta]`
`=2[(a+b)^(2)-(a^(2)+b^(2))]=2(2ab)=4ab`
`P=(alpha+beta)^(2).(alpha-beta)^(2)=(alpha+beta^(2))[(alpha+beta)^(2)-4alphabeta]`
`=(a+b)^(2)[(a+b)^(2)-2(a^(2)+b^(2))]`
`=(a+b)^(2)(a^(2)+b^(2)+2ab-2a^(2)-2b^(2))`
`=(a+b)^(2)(2ab-a^(2)-b^(2))`
`=-(a+b)^(2)(a^(2)+b^(2)-2ab)`
`=-(a+b)^(2)(a-b)^(2)=-(a^(2)-b^(2))^(2)`
`:. ` Required polynomial is, `k(x^(2)-Sx-P)`
`=k[x^(2)-4ab.x-(a^(2)-b^(2))^(2)]" "` Ans.