Find the value of this expression (Canadian Math Olympiad, 1996)?

Question

Let a, b, and c be the roots of x^3-x-1=0.

Calculate the value of

(1+a)/(1-a)+(1+b)/(1-b)+(1+c)/(1-c).

Mathematically justify your answer, plz :)

Answer 1

a, b, c are roots of x³ - x - 1 = 0

=> (x-a)(x-b)(x-c) = x³ - x - 1

=> x³ -(a+b+c)x² + (ab+bc+ca)x - abc = 0

=> a+b+c = 0, ab+bc+ca = -1 and abc = 1

(1+a)/(1-a)+(1+b)/(1-b)+(1+c)/(1-c).

= [(1+a)(1-b)(1-c) + (1+b)(1-c)(1-a) + (1+c)(1-a)(1-b)] / (1-a)(1-b)(1-c)

= [3 -(a+b+c) - (ab+bc+ca) +3abc] / [1 - (a+b+c) + (ab+bc+ca) -abc]

= [3 - 0 + 1 + 3] / [1 - 0 -1 - 1]

= - 7.