Theorems on the Roots of Polynomial Equations
If p(x) and d(x) are any two nonconstant polynomials then there are unique
polynomials q(x) and R(x) such that
p(x) = d(x) q(x) + R(x)
where R(x) is either zero or it is of lower degree than d(x).
Remainder Theorem: If the polynomial p(x) is divided by (x – r) then
the constant remainder R
is given by R = p (r).
Factor Theorem: The number c is a solution of the polynomial equation
p (x) = 0
if and only if p (x) has (x – c) as a factor.
Complete Factorization Theorem:
then there are n numbers:
not necessarily distinct,
The ci‘s are the zeros of p(x) and they may be either real or complex
numbers. The complex zeros always occur in conjugate pairs whenever
all coefficients of p (x) are real numbers.
Fundamental Theorem of Algebra:
Every nonconstant polynomial has at least one (real or complex) zero.
Real Factors Theorem:
Any polynomial with real coefficients can be factored into a product of linear
and quadratic polynomials having real coefficients, where the quadratic
polynomials have no real zeros.
Rational Roots Theorem:
have all integral coefficients. If c/d is a rational solution, in reduced form,
then c divides ao exactly and d divides an exactly.
Descartes’s Rule of Signs:
Suppose that p(x) is a polynomial with real coefficients and with terms written
in descending powers of the variable. Then
(i) the number of positive roots to p(x)=0 is either equal to N: the number
of variations in sign in the coefficients of p(x), or else it is less than N by
an even integer.
(ii) the number of negative roots to p(x)=0 is either equal to M: the number
of variations in sign in the coefficients of p(–x), or else it is less than M by
an even integer.
Upper and Lower Bound Theorem for Real Roots:
Suppose the polynomial:
has all real coefficients and an>0. Then
(i) the positive number B is an upper bound on the real roots to p(x)=0 if
and the numbers are all nonnegative.
(ii) the negative number b is a lower bound on the real roots to p(x)=0 if
and the sequence of numbers are alternately
nonpositive and nonnegative or vice versa.
Location Theorem: If p(x) is a polynomial with real
coefficients and a,b are real numbers with
a < b and p(a) and p(b) have opposite sign then there exists at least one
number c such that a < c < b and p(c) = 0.