Question:
If ,
,
be the roots of the cubic equation
. Prove that the equation in y whose roots are
is obtained by the transformation
. Hence, form the equation with above roots.
Solution:
Given that ,
,
are the roots of the equation:
…call this equation I.
By relationships between roots and co-efficients, (Viete’s relations), we get
and
, and
Now, , that is,
, or
…call this equation II.
Subtracting Equation II from Equation I, we get
since
which is the required transformation.
Now, , that is,
Putting this value of x in Equation I, we get
, that is,
, which is the required equation.
Cheers,
Nalin Pithwa.