Question

If `alpha and beta` are the zeros of the quadratic polynomial `f (x)= x^2-2x+3`, find a polynomial whose roots are (i) `alpha-2,beta-2` (ii) `(alpha-1)/(alpha+1),(beta-1)/(beta+1)`

Answer 1

1)`x^2-2x+3`
`alpha+beta=2`
`alpha*beta=3`
`x^2-((alpha-2)(+(beta-2))x+(alpha-2)(beta-2)`
`alpha-2+beta-2=(alpha+beta-4)`
`(alpha-2)(beta-2)=alphabeta-2beta-2alpha+4`
`3-2(2)+4=3`
`x^2+2x+3=0`.
2)`(alpha-1)/(alpha+1)+(beta-1)/(beta+1)`
`=((alpha-1)(beta+1)+(beta-1)(alpha+1))/((alpha+1)beta+1)`
`=(2alphabeta-2)/(alphabeta+alpha+beta+1)`
`=(2*3-2)/(3+2+1)=4/6=2/3`
`=((alpha-1)/(alpha+1))((beta-1)/(beta+1))`
`=(alphabeta-beta-alpha+1)/(alphabeta+alpha+beta+1)`
`=(3-(2)+1)/(3+2+1)=2/6=1/3`
`=x^2-2/3x+1/3=0`
`3x^2-2x+1=0`.