Question

if alpha and beta are the zeroes of (x)(x) -5x+6, find polynomial whose zeroes are (1) 2alpha-1 and2beta-1

Answer 1

let 2 roots be a and b  of  x² - 5x + 6 = 0

sum of roots = a + b = - (-5) / 1 = 5
products of roots = ab = 6/1 = 6


now new roots are  2a - 1  and  2b - 1

sum of new roots =  2a -1 + 2b - 1 = 2(a + b) - 2 = 2(5) - 2 = 8

product of new roots = (2a -1)(2b-1) = 4ab - 2a - 2b +1 = 4ab -2(a + b) + 1

                                                                 = 4(6) -2(5) + 1 = 15

so new quadratic polynomial would be = x² - 8x + 15 = 0

hope this helps.