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162 ALGEBRA

Example 1. The equation
5
2
4
3
6
ax + bx + cx + dx + cx + bx + a = 0
can be reduced to the cubic equation
3
2
ay + by +(c – 3a)y + d – 2b = 0
by the substitution y = x + 1/x.

5.1.5-2. Equations of general form and their properties.

An algebraic equation of degree n has the form

n
a n x + a n–1 x n–1 + ··· + a 1 x + a 0 = 0 (a n ≠ 0), (5.1.5.1)

where a k are real or complex coefficients. Denote the polynomial of degree n on the
right-hand side in equation (5.1.5.1) by

n
P n (x) ≡ a n x + a n–1 x n–1 + ··· + a 1 x + a 0 (a n ≠ 0). (5.1.5.2)

Avalue x = x 1 such that P n (x 1 )= 0 is called a root of equation (5.1.5.1) (and also
a root of the polynomial P n (x)). A value x = x 1 is called a root of multiplicity m if
m
P n (x)= (x–x 1 ) Q n–m (x), where m is an integer (1 ≤ m ≤ n), and Q n–m (x) is a polynomial
of degree n – m such that Q n–m (x 1 ) ≠ 0.

THEOREM 1(FUNDAMENTAL THEOREM OF ALGEBRA). Any algebraic equation of degree
n has exactly n roots (real or complex), each root counted according to its multiplicity.

Thus, the left-hand side of equation (5.1.5.1) with roots x 1 , x 2 , ... , x s of the respective
multiplicities k 1 , k 2 , ... , k s (k 1 + k 2 + ··· + k s = n) can be factorized as follows:

k 1
k 2
P n (x)= a n (x – x 1 ) (x – x 2 ) ... (x – x s ) .
k s
THEOREM 2. Any algebraic equation of an odd degree with real coefficients has at least
one real root.
THEOREM 3. Suppose that equation (5.1.5.1) with real coefficients has a complex root
x 1 = α + iβ. Then this equation has the complex conjugate root x 2 = α – iβ, and the roots
x 1 , x 2 have the same multiplicity.

THEOREM 4. Any rational root of equation (5.1.5.1) with integer coefficients a k is an
irreducible fraction of the form p/q,where p is a divisor of a 0 and q is a divisor of a n .If
a n = 1, then all rational roots of equation (5.1.5.1) (if they exist) are integer divisors of the
free term.

THEOREM 5(ABEL–RUFFINI THEOREM). Any equation (5.1.5.1) of degree n ≤ 4 is
solvable by radicals, i.e., its roots can be expressed via its coefficients by the operations
of addition, subtraction, multiplication, division, and taking roots (see Subsections 5.1.2–
5.1.4). In general, equation (5.1.5.1) of degree n > 4 cannot be solved by radicals.