Question

If alpha and beta are zeroes of quadratic polynomial f(x)=x square-3x+3. Find a polynomial whose zeroes are alpha+2 and beta+2?

Answer 1

α and β are the zeros of 3x²-4x +1 polynomial,
first of all we factorise 3x²-4x+13x² -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 = 1or α = 1 and β. = 1/3you 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