Question

If α and β are zeros of x
^2- x - 2, find a polynomial whose zeros are (2α + 1) and (2β + 1).​

Answer 1

Answer:

x² - 4x -5

Step-by-step explanation:

This question can be solved using two methods.

METHOD I :

If α & β are zeroes of f(x) =x²-x-2

then

α + β = -b/a = 1

α β = c/a = -2

let α =2α+1 & β = 2β+1

α + β = 2α+1 + 2β+1

=2α+2β+2 =2(α+β)+2 =2(1)+2=4

α × β=(2α+1)(2β+1) = 4α β+2α+ 2β+1 = 4(-2)+2(1)+1=-8+2+1=-5

Polynomial having α & β as zeroes is given by

k (x²-(α’ + β’)x +α’ β’)

= k (x² - 4x -5)

By giving different values to k, there can be infinite polynomials

Hence, the required polynomial is

x² - 4x -5.

METHOD II :

f(x) =x²-x-2 =x² -2x + x -2 = x(x-2) +1(x-2) =(x+1)(x-2)

so two zeroes are -1 and 2

α = -1 & β= 2

α’ = 2α + 1 = 2(-1)+1 = -1

β’ = 2β +1= 2(2)+1=5

Polynomial having α’ & β’ as zeroes is given by

k (x²-(α’ + β’)x +α’ β’)

k (x²-(-1 + 5)x +(-1)(5))

= k (x² - 4x -5)

Hence, the required polynomial is

x² - 4x -5.