Question

If alpha and beta are the zeros of the quadratic polynomial 3x^2-4x+1, find a quadratic polynomial whose zeros will be alpha^2/beta and beta^2/alpha.

Answer 1

α and β are the zeros of 3x²-4x +1 polynomial,

first of all we factorise 3x²-4x+1
3x² -4x + 1

=3x² -3x -x +1

=3x( x -1) -1(x -1)

=(3x -1)(x -1)

hence. (3x -1) and (x -1) are the factors of given polynomial .

so, x = 1/3 and 1 are the zeros of that polynomial.

hence, α = 1/3. and β = 1
or α = 1 and β. = 1/3
you can choose any one in both

I choose α = 1. and β = 1/3

now,
let any unknown. polynomial. whose zeros are α²/β and β²/α

α²/β = (1)²/(1/3) = 3

β²/α = (1/3)²/1 = 1/9

now, equation of unknown polynomial.

x²- ( sum of roots)x + product of roots

= x²- ( α²/β + β²/α)x +(α²/β)(β²/α)

put α²/β = 3 and β²/α = 1/9

= x²- ( 3 +1/9)x + 3 × 1/9

= x² -28x/9 + 3/9

={ 9x² -28x + 3 }1/9

hence, 9x² -28x + 3 is answer

Answer 2

I hope you understand