Question

If alpha and beta are the zeros of x^2-2x-1 form a polynomial whose zeroes are 2alpha -1,2beta-1

Answer 1

Let alpha = a and beta = b

Sum of zeroes of polynomial = -b/a

a + b = 2

Product of zeroes = c/a

ab = -1

Roots of the new polynomial = 2a-1 , 2b-1

Sum of zeroes = 2a - 1 + 2b -1

= 2a +2b - 2 = 2( a + b - 1 )

As a+ b = 2

= 2 ( 2- 1 ) = 2 (1) = 2 = -b/a

Product of zeroes = (2a-1) (2b -1)

= 2ab - 2a -2b +1

As ab = -1 and a+ b = 2

= 2 (-1) - 2 (2) +1

= -2-4+1 = -6+1 = -5 = c/a

So polynomial is x^2 -2x -5