Question

if alpha beta are the zeros of x square - 2 x minus one form a quadratic polynomial whose zeros are 2 alpha minus 2 beta minus one

Answer 1

Answer:

Do you mean the roots should be 2α-1 and 2β-1?  If so, the answer is...

x² - 2x - 7

Step-by-step explanation:

Since α and β are roots of x² - 2x - 1, we have

α + β = 2  and  αβ = - 1.

Let the polynomial we're looking for be x² - Ax + B.

Since the roots are 2α-1 and 2β-1, we must have

A = ( 2α - 1 ) + ( 2β - 1 ) = 2 ( α + β ) - 2 = 2 ( 2 ) - 2 = 4 - 2 = 2

and

B = ( 2α - 1 ) ( 2β - 1 ) = 4 α β - 2 ( α + β ) + 1 = 4 ( -1 ) - 2 ( 2 ) + 1 = - 4 - 4 + 1 = -7.

So the quadratic is

x² - Ax + B = x² - 2x - 7