# 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

then there are n numbers:
not necessarily distinct,

such that

The c_{i}‘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

have all integral coefficients. If c/d is a rational solution, in reduced form,

then c divides a_{o} exactly and d divides a_{n} 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 a_{n}>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.