1. 2.5 Zeros of Polynomial Functions

2. If the f(x) has any rational zeros they must be on this list
Rational Zeros Theorem: Let f be a polynomial function of degree 1 or higher of the form
where each coefficient is an integer. If p/q in lowest term is a factor of f then p is a factor of a0 and q is a factor an ?
If the f(x) has any rational zeros they must be on this list
The integer factors of -7 are p
The integer factors of 3 are q
f(x) may not have any rational zeros and can have at most 3 because it is 3rd degree so not all the numbers on the list can be zeros
If I make a list of all the possible p/q’s it will be

3. There is actually one rational zero 7/3
The other two zeros are irrational
7/3

4. Make a list of the possible rational zeros according to the rational zeros theorem
Reduce the fractions and get rid of doubles

5. List the possible rational zeros
Find the zeros of
The equation is 3rd degree so it can have at most 3 zeros
List the possible rational zeros
Graph and find one of the numbers on the rational zeros list where the graph the graph crosses the x-axis
Remember the calculator makes rounding errors 0 on the calculator comes up as 1.23 E -12
You can also use to find x-intercepts.
If you don’t have a graphing calculator you can plug the values in until one equals zero or synthetically divide them until you get a remainder of zero.

6. 1 5 -1 -2 2 -5 Find the zeros of List the possible rational zeros
The equation is 3rd degree so it can have at most 3 zeros
List the possible rational zeros
Graph and find one of the numbers on the rational zeros list where the graph the graph crosses the x-axis
Y-max= 10
The graph appears to cross at x = -1 so x+1 must be a factor. Use synthetic division to divide .
X-max= 4
X-min= -5 1 5 -1 -2 2 -5
Y-min= -10
The zeros are
Use the quadratic formula to solve the quadratic

7. List the possible rational zeros
The equation is 3rd degree so it can have at most 3 zeros
Find the zeros of
List the possible rational zeros
Graph and find one of the numbers on the rational zeros list where the graph the graph crosses the x-axis

8. 1 2 -12 20 -6 10 Find the zeros of List the possible rational zeros
The equation is 3rd degree so it can have at most 3 zeros
Find the zeros of
List the possible rational zeros
Graph and find one of the numbers on the rational zeros list where the graph the graph crosses the x-axis
y-max= 12
The graph appears to only cross at x = 2 so x-2 must be a factor. Use synthetic division to divide . 1
X-min= -1
X-max= 6 2
-12 20 -6 10
y-min= -12
The zeros are
Since two of the zeros are imaginary the graph only crosses the x-axis once.
Use the quadratic formula to solve the quadratic

9. 3 2 -4 -6 The graph looks like it may cross at 2/3 ,-4/3 & 4/3
Find the zeros.
List the possible rational zeros.
The graph looks like it may cross at 2/3 ,-4/3 & 4/3
Using the trace feature it only crosses at 2/3 3 2 -4 -6
The zeros are

10. Solve over the complex number system:
The graph only crosses the x-axis once so there is one real root and two complex roots that are not real
13/7
Lets make the list of possible rational roots

11. Solve over the complex number system:
13/
13/7
The solutions are

12. Solve for exact solutions:
P #3, 5, 8 List possible rational zeros
#9-15 odd
List possible rational zeros
Use the list and the graph to find an actual zero
Use synthetic division to factor the polynomial
Factor or use the quadratic formula on the remaining quadratic to find the other zeros.
#19
Solve for exact solutions:

13. Find an exact value for the zeros of the polynomial function
There can be at most 4 real zeros because f(x) is 4th degree
The list of possible rational zeros is
Y-max= 20
Next graph the equation 3 -1
X-min= -6
X-max= 6
Hit trace and plug in each number in the list of possible rational zeros to find which are really zeros
3 and -1 are the only rational zeros the other two are irrational
Y-min= -60

14. Find the zeros
3 and -1 are the only rational zeros the other two are irrational
Factor the equation by using synthetic division -1
Now divide -1 into the 3rd degree polynomial

15. -1 is a zero
3 is a zero
The other two irrational zeros come from this factor. To find where it equals zero use the quadratic formula
Solve using the quadratic formula
Use trace on your calculator to make sure your zeros are correct
The exact real zeros are

16. The graph crosses at 5 & -5 1 -1 -29 25 100 5 20 -45 -100 1 4 -9 -20 0
Find the zeros
List possible rational zeros
The graph crosses at 5 & -5
Y-max= 160
X-max= 10
X-min= -10 -5
Y-min= -160

17. Find the zeros

18. Solve over the complex number system:
List the possible rational roots
The graph appears to touch at -3 therefore it must be a zero of even multiplicity 2

19. Solve over the complex number system:
Since the graph just touches at -3 it is a zero of even multiplicity and the factor (x+3) occurs more than once so we will need to do synthetic division with it at least twice. -3

20. There are roots are -3, -1+i, -1-i
Solve over the complex number system.
Use the quadratic formula
There are roots are -3, -1+i, -1-i
You can test these by plugging them back in the equation on your calculator

21. Find a second degree polynomial that has zeros of 3 and -4, in addition f(0)=24

22. Find a third degree polynomial that has zeros of 2, -3 and 1, in addition f(3)=12

23. Find a third degree polynomial that has zeros of 2i and 1, in addition f(2)=24
If 2i is a zero its conjugate must also be a zero

24. If 2+i is a zero 2-i is also a zero
Find a 3rd degree polynomial f with real coefficients with zeros -5 & 2+i where f(2)=14
If 2+i is a zero 2-i is also a zero

25. P 357
22-24
25, 27, 42, 46
List possible rational roots
Use the list and the graph to find an actual root
Use synthetic division to factor the polynomial
Factor or use the quadratic formula on the remaining quadratic to find the other roots.