Question

If Alpha and beta are zeros of polynomial P(X) =3x^2-4x+1, find a cubic polynomial whose zeros are zer find a cubic polynomial whose zeros are 0,alpha ^2/beta, beta^2/alpha

Answer 1

The given quadratic equation is
3x²-4x+1=0
or,3x²-3x-x+1=0
or,3x(x-1)-(x-1)=0
or,(3x-1)(x-1)=0
or,x=1/3 and 1 aare zeros of this polynomial.
alpha=1/3 and beta=1
Now we have to determine a cubic polynomial with roots 0,alpha²/beta and beta²/alpha.
so,alpha²/beta=1/9
and beta²/alpha=3
Hence the cubic polynomial is
(x-0)(x-1/9)(x-3)
or,x(x²-28x/9+1/3)
or,x³-28x²/9+x/3=0. (1)
is one of the cubic polynomials.You could find another polynomials by multiplying eqn (1) by any number(ex-multiply it by 9).