Question

, are zeros of x² - 4x - 3, find a quadratic polynomial whose zeros are 3α and 3β

Answer 1

Given  α and β are zeroes of the polynomial f(x) = x2- 4x + 3

α+ β = 4 αβ = 3

1) (3α + 3β) = 3x 4 = 12

3α x 3β = 9 x 3  =  27.

If 3α, 3β are zeros of the quadratic polynomial then the equation is

 x2 -(3α + 3β)x + 9αβ = 0 then

x2 - 12x + 27 = 0.

2)
(1/2α + 1/2β) = (α + β) / 2αβ  = 4 / 6 = 2 / 3.

1/4αβ  = 1 /12

If 1 / 2α, 1 / 2β are zeros of the quadratic polynomial then the equation is

 x2 -(1 / 2α + 1 / 2β)x + 1 / 4αβ = 0 then

x2 -(2 / 3)x + 1 / 12 = 0

12x2 - 8x + 1  = 0.