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
Equation
# of Roots
2ix  5 x  (3  2i ) x  7  0
3
3x  11x  19 x  25 x  36  0
4
3
4
2
3
2
2.7 General Results for
Polynomial Equations b 
x
b2  4ac
2a
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
2.7 General Results for
Polynomial Equations
sum
product
Equation
an 1

an
2 x3  5 x 2  3x  9  0
5/ 2
a0

if n is odd
an
9 / 2
a0
if n is even
an
x4  x2  x  3  0
0
3
This rule is extremely useful with quadratics.
Theorems
2.7 General Results for
Polynomial Equations
General Quadratic Equation:
an 1
b
Sum of the roots  
 
ax  bx  c  0
a
an
c
a0

Product of the roots =
an a
General Form of a Quadratic!
2
ax  bx  c  0 
2
a 2  b 
c 0
x   x 
a
a a
 a 
 x2   sum of the roots  x   product of the roots   0
2.7 General Results for
Polynomial Equations
 x2   sum of the roots  x   product of the roots   0
Find a quadratic equation with roots 2  3i
sum =  2  3i    2  3i   4
product =  2  3i  2  3i   13
x  4 x  13  0
2
Find a cubic equation with integral coefficients that has no
quadratic term and 3  i 2 is one of the roots.
an 1
0
sum = 
 0
an
a3
a3 x  0 x  a1 x  a0  0
3
2
3  i 2 is one of the roots 
let r  3rd root
3  i 2 is also a root
sum  0   3  i 2    3  i 2   r  0
r  6
So the linear factor of P( x)  x  6.
Quadratic Factor: Use 3  i 2
x   sum x  product
2


x2  6x  3  i 2 3  i 2
x2  6x  9  2

x  6 x  11
2
 cubic equation is...
 x  6   x 2  6 x  11  0
2.7/1-25ODD