Question

If a and b are the roots of the polynomial f(x) = x² – 2x + 3, find the polynomial whose roots are

(i) α + 2, β + 2

(ii) (α - 1)/(α + 1), (β - 1)/(β + 1) 

Answer 1

(i) α + 2, β + 2 are the roots (given)

Sum of the roots = α + 2 + β + 2

= α + β + 4

= 2 + 4 = 6

Product of the roots = (α + 2) (β + 2)

= αβ + 2α + 2β + 4

= αβ + 2(α + β) + 4

= 3 + 2 x 2 + 4

= 3 + 4 + 4 = 11

∴ The required equation = x² – 6x + 11 = 0.

(ii) (α - 1)/(α + 1), (β - 1)/(β + 1)

⇒ 3x² – 2x + 1 = 0