Question

I have the polynomial f=1+x-2x^2+x^3 from C[x] with the roots z1,z2,z3.
I need to form the polynomial with the roots: y1=1-z1, y2=1-z2 and y3=1-z3.

Answer 1

your new polynomial will look like this: \[a X^{3} + b X^{2} + c X + d = 0\]
and you find it using the formula:\[X^{3} - S_{1}X^{2} + S_{2}X - S_{3} = 0\]
where \[S_{1} = y_{1}+ y_{2} + y_{3}\] \[S_{2} = y_{1}y_{2} + y_{1}y_{3} + y_{2}y_{3}\]
\[S_{3} = y_{1}y_{2}y_{3}\]