Download 2.7 General Results for Polynomial Equations

2.7 General Results for
Polynomial Equations
THM 1. The Fundamental Theorem of Algebra
In the complex number system consisting of all real and
imaginary numbers...
P( x) is a polynomial of degree n
(n  0) with complex coefficients

P( x) has exactly n roots (provided
a double root is counted as two roots,...)
2.7 General Results for
Polynomial Equations
THM 2. Complex Conjugates Theorem
a  bi is an imaginary root of P ( x)  a  bi is also a root
THM 3.
Suppose P( x) is a polynomial with rational coefficients, and
a and b are rational numbers such that b is irrational. If
a  b is a root of P( x), then a  b is also a root.
2.7 General Results for
Polynomial Equations
Table
THM 4.
P( x) is a polynomial of odd degree with real coefficients
 P( x) has at least one real root.
THM 5.
an x n  an 1 x n 1  an 2 x n  2  ...  a0  0, an  0
an 1
the sum of the roots is 
;
an
a0
the product of the roots is
{
an
if n is even
a0

if n is odd
an
Table