Download Theorems on the Roots of Polynomial Equations

Theorems on the Roots of Polynomial Equations
Division Algorithm:
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:
If
p (x) = an xn + an-1 xn-1 + . . . + a1 x + ao ,
an ≠ 0,
then there are n numbers: c1 , c2 , . . . , cn , not necessarily distinct,
such that
p (x) = an (x – c1)(x – c2) . . . (x – cn) .
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:
Let
an xn + an-1 xn-1 + . . . + a1 x + ao = 0
have all integral coefficients. If
c
is a rational solution, in reduced form,
d
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:
p(x) = an xn + an-1 xn-1 + . . . + a1 x + ao
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
p(x) = (x – B)(Bn-1 xn-1 + Bn-2 xn-2 + . . . + B1 x + Bo) + R1
and the numbers Bn-1 , Bn-2 , . . . , B1, Bo , R1 are all nonnegative.
(ii) the negative number b is a lower bound on the real roots to p(x)=0 if
p(x) = (x – b)(bn-1 xn-1 + bn-2 xn-2 + . . . + b1 x + bo) + R2
and the sequence of numbers bn-1 , bn-2 , . . . , b1, bo , R2 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.