Download 2.7 General Results for Polynomial Equations

Use the Rational Root THM to determine algebraically
where the two graphs intersect or are tangent.
y  x3  3x
y2
 x3  3x  2
x3  3x  2  0
p  1, 2
q  1
p / q  1, 2
1 0 -3 -2
1
1 1 -2
1 1 -2 -4
1 0 -3 -2
1 -1 1 2
1 -1 -2 0
x 2  x  2  0   x  2  x  1  0
x  2, 1  1 is a double root 
 1, 2  &  2, 2 
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
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:
ax  bx  c  0
2
an 1
Sum of the roots  
 b
a
an
a0
c
Product of the roots =

an
a
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  0
2



x 2  6 x  11  0
 cubic equation is...
 x  6   x 2  6 x  11  0
x  6 x  3+i 2 3  i 2  0
3
2
2
2
x

6
x

11
x

6
x
 36 x  66  0
x  6x  9  2  0
2
x3  25 x  66  0
2.7 General Results for
Polynomial Equations
or




 x  3  i 2   x  3  i 2   x   6    0




2.7/1-25ODD
Find the sum and the product of the roots of
3x  5 x  x  2  0
3
2
THM 5.
an x  an 1 x
n 1
 an 2 x
n2
 ...  a0  0, an  0
an 1
the sum of the roots is 
;
an
a0
n
the product of the roots is
{
Sum  5 / 3
Product  2 / 3
an
if n is even
a0

if n is odd
an