Page 196 -

5.1. POLYNOMIALS AND ALGEBRAIC EQUATIONS 163

5.1.5-3. Relations between roots and coefficients. Discriminant of an equation.

VI` ETE THEOREM. The roots of equation (5.1.5.1) (counted according to their multiplicity)
and its coefficients satisfy the following relations:

(–1) k a n–k = S k (k = 1, 2, ... , n),
a n
where S k are elementary symmetric functions of x 1 , x 2 , ... , x n :
n n n

S 1 = x i , S 2 = x i x j , S 3 = x i x j x k , ... , S n = x 1 x 2 ... x n .
i=1 1≤i<j 1≤i<j<k

Note also the following relations:
k

(n – k)a n–k + a n–(k–j) s j = 0 (k = 1, 2, ... , n)
j=1
n
j
with symmetric functions s j = x .
i
i=1
2n–2
The discriminant D of an algebraic equation is the product of a n and the squared
Vandermonde determinant Δ(x 1 , x 2 , ... , x n )ofits roots:
2n–2 2 2n–2 2
D = a n [Δ(x 1 , x 2 , ... , x n )] = a n (x i – x j ) .
1≤j<i≤n
The discriminant D is a symmetric function of the roots x 1 , x 2 , ... , x n , and is equal to zero
if and only if the polynomial P n (x) has at least one multiple root.

5.1.5-4. Bounds for the roots of algebraic equations with real coefficients.
1 . All roots of equation (5.1.5.1) in absolute value do not exceed
◦
A
N = 1 + , (5.1.5.3)
|a n |
where A is the largest of |a 0 |, |a 1 |, ... , |a n–1 |.
The last result admits the following generalization: all roots of equation (5.1.5.1) in
absolute value do not exceed
A 1
N 1 = ρ + , (5.1.5.4)
|a n |
where ρ > 0 is arbitrary and A 1 is the largest of
|a n–2 | |a n–3 | |a 0 |
|a n–1 |, , 2 , ... , n–1 .
ρ ρ ρ
For ρ = 1, formula (5.1.5.4) turns into (5.1.5.3).
Remark. Formulas (5.1.5.3) and (5.1.5.4) can also be used for equations with complex coefficients.
Example 2. Consider the following equation of degree 4:
2
4
P 4(x)= 9x – 9x – 36x + 1.
Formula (5.1.5.3) for n = 4, |a n| = 9, A = 36 yields a fairly rough estimate N = 5, i.e., the roots of the equation
belong to the interval [–5, 5]. Formula (5.1.5.4) for ρ = 2, n = 4, |a n| = 9, A 1 = 9 yields a better estimate for
the bounds of the roots of this polynomial, N 1 = 3.