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FAMOUS PROBLEMS OF MATHEMATICS
equation. If, I and, 2 are the roots of x 2 + bx + c = 0, then , I + '2 = -b and, 1-'2 = c. In general it can be proven that if'l> '2' '3' "', 'n are the roots of the equationxn + alxn- I + a 2 x n- 2 + ... + an-IX + an = 0, then t'I=-a.
1=.
t t
I.J=I
I,J,k= I
'1' J = a 2 'I'J'k
(i < j)
= -a 3
(i < j, i
< k,j < k)
1: '1' J means the sum of all terms formed by multiplying any two roots together.
PROBLEM SET VII-C 1. If the roots of x 3
+ a l x 2 + a 2x + a 3 =
0 are 'I' '2' 'J' then we can write the equation in the form (x - , I) -(x - , 2)(X - , 3) = O. By multiplying the factors of the left-hand side of the equation, show that
'I +'2 +'3 =
-a l
'1'2 + '1'3 + '2'3 = '.'2'3 = -aJ
a2
2. Repeat the process for the equation x 4 + alx J + a 3 x + a4 = O.
+ a 2x 2