Unit 2
When the is a factor of
the .
Another way too say it
When is a of the
polynomial then .
The Factor Theorem
Remember the Fundamental Theorem of Algebra
A polynomial of degree has
exactly .
Remember that the degree tells you what …..
•
•
•
If is a zero of the function , then
o
is a or root of the
polynomial equation .
o
is a of the
polynomial .
o
is an of the graph
Complex numbers
Can real numbers be complex?
A of a real and an number of the form
Conjugate Pairs
Are binomial where the
Always in !
Degree
Zeros
Possible Combinations
5
Extend...
# of zeros:
# of Real Zeros:
# of Imaginary Zeros:
Determine the
number of zeros
for the
polynomials.
Then graph the polynomial.
Identify
the number of real zeros and the number of
imaginary zeros.
# of zeros:
# of Real Zeros:
# of Imaginary Zeros:
Determine the
number of zeros
for the
polynomials.
Then graph the polynomial.
Identify
the number of real zeros and the number of
imaginary zeros.
Example 2:
# of zeros:
# of Real Zeros:
# of Imaginary Zeros:
Determine the
number of zeros
for the
polynomials.
Then graph the polynomial.
Identify
the number of real zeros and the number of
imaginary zeros.
# of zeros:
Determine the
number of zeros
for the
polynomials.
Then graph the polynomial.
Identify
Example 5:
# of zeros:
Determine the number of zeros.
Then find the
real and imaginary solutions for each equation.
Check using Technology. (Need to factor!!)
Example 6:
# of zeros:
Example 7:
# of zeros:
Determine the number of zeros.
Then find the
real and imaginary solutions for each equation.
Check using Technology. (Need to factor!!)
Example 8:
# of zeros:
Example 7a:
# of zeros:
Determine the number of zeros.
Then find the
real and imaginary solutions for each equation.
Check using Technology. (Need to factor!!)
Example 8a:
# of zeros:
Determine the number of zeros.
Graph the following equations on the same grid.
The previous graph will generate three different equations.
1. 2. 3.
Which of the above equations would you prefer to solve.
Graph
So if I see an equation like
What do we know? What can we figure out?
Ways to Solve
1.
Graph .
2.
Set Equal to .
Find all the solutions to the following polynomial
equations. Check using Technology.
Example 9:
# of zeros:
Find all the solutions to the following polynomial
equations. Check using Technology.
Example 10:
# of zeros:
Hint: Get equations equal to zeros
Find all the solutions to the following polynomial
equations. Check using Technology.
Example 11:
# of zeros:
Find all the solutions to the following polynomial
equations. Check using Technology.
Example 12:
# of zeros:
Hint: Get equations equal to zeros
Where are the zeros of
?
a)
b)
c)
d)
How to Solve...