Algebraic equations of the n-th degree, multiple roots, number of roots of an algebraic equation of the n-th degree.
Algebraic functions are functions whose values, for a given value of
x, are calculated using only arithmetic operations (addition, subtraction, multiplication and division) and operations of raising to a power (including with a rational exponent).
An algebraic equation of degree
n with one unknown
x is an equation
P(x)=0, where
P(x) algebraic function of the form
P(x)=a0xn + a1xn-1 + ... + an-1x + an,
the coefficients
ai are real (or complex) numbers, and
a0 ≠ 0.
The algebraic expression
P(x) is called a polynomial of degree
n with respect to
x,
the coefficient
an is a free member of the polynomial.
To solve an equation means to find the values of
x for which the equality
P(x)=0 holds.
The found values of
x are called the roots of the equation or the zeros of the function
P(x).
If the equality
P(x)= f(x)(x-x0)m is satisfied, where f(x) is a polynomial and
f(x0) ≠ 0,
then the root
x=x0 is a root of multiplicity
m.
The fundamental theorem of polynomial algebra states that an algebraic equation of degree
n has exactly
n roots
if a root of multiplicity
m is counted
m times.
A complete solution of an algebraic equation consists in finding all the roots with their multiplicities.
Only for algebraic equations of the first degree (linear), second degree (quadrate), third degree (cubic) and fourth degree (quartic),
there are formulas expressing the roots of these equations through their coefficients using a finite number of arithmetic operations
(addition, subtraction, division, multiplication) and root extractions. There are no such formulas for equations of higher degrees.
Linear equations with one variable
Quadratic equations
Cubic equations
Quartic equations
Equations of higher degrees
