1. 2.5 Zeros of Polynomial Functions
Fundamental Theorem of Algebra
Rational Zero Test
Upper and Lower bound Rule

2. Fundamental Theorem of Algebra
If f(x) is a polynomial of degree “n” > 0, then f(x) has at least one zero in the complex number system.
Complex zero’s (roots) come in pairs
If a + bi is a zero, then a – bi is a zero.

3. Linear Factorization Theorem
If f(x) is a polynomial of degree “n”>0, then there are as many zeros as degree.
If f(x) is a third degree function, then
f(x) = an(x – c1)(x – c2)(x – c3) where c are complex numbers.
Complex zero’s (roots) come in pairs
If a + bi is a zero, then a – bi is a zero.

4. The Rational Zero Test
If f(x) has integer coefficients, then all possible zeros are
factors of the constant
factor of the lead coefficient

5. The Rational Zero Test
If f(x) has integer coefficients, then all possible zeros are
factors of the constant
factor of the lead coefficient
f(x) = x 3 – 7x 2 + 4x + 12
Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± 12
± 1

6. How do you want to find the other zeros. x 2 – 8x + 12
f(x) = x 3 – 7x 2 + 4x + 12 Possible zeros ± 1, ± 2, ± 3, ± 4, ± 6, ± ± 1
- 1 |
So – 1 is a zero
How do you want to find the other zeros.
x 2 – 8x + 12

7. Find the zeros f(x) = 3x3 – x2 + 6x - 2

8. Descartes' Rule of Signs
Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with real coefficients and a0 ≠ 0.
Part 1
The number of positive real zeros equals (or a even number less), the number of variation in the sign of the coefficient (switching from positive to negative or negative to positive).

9. Descartes' Rule of Signs
Let f(x) = anxn + an-1xn-1 + ….a1x + a0 ; with real coefficients and a0 ≠ 0.
Part 2
The number of negative real zeros equals (or a even number less), the number of variation in the sign of the coefficient
(switching from positive to negative or negative to positive) in f(- x).

10. Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
How many times does the sign change ?

11. Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
How many times does the sign change ?
3 times.
There are 3 or 1 positive zeros.

12. Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
What about f( -x) = -4x3 – 3x2 – 2x - 1
How many times does the sign change ?

13. Using the Desecrate rule of signs
f(x) = 4x3 - 3x2 +2x – 1
What about f( -x) = -4x3 – 3x2 – 2x - 1
How many times does the sign change ?
No change, no negative zeros.

14. Upper and Lower bound Rule
If c > 0 ( “c” the number you divide by) and the last row of synthetic division is all positive or zero, the c| is the upper bound
So there is no zero larger then c, where c > 0.
If c < 0 and the last row alternate signs
( zero count either way), then c is the lower bound.

15. f(x) = 2x3 – 5x2 + 12x - 5 Check to see if 3 is the upper bound?
3| All signs are positive.
3 is an upper bound

16. f(x) = 2x3 – 5x2 + 12x - 5 Check to see if - 1 is the lower bound?
- 1| All signs are switch.
-1 is an lower bound

17. f(x) = 2x3 – 5x2 + 12x - 5
Find the zeros

18. Homework
Page 160 – 164
# 5, 15, 23, 35,
42, 50, 57, 65,
73, 81, 85, 93,
103, 108, 111

19. Homework
Page 160 – 164
# 9, 19, 29, 41,
53, 61, 64, 77,
87, 97, 105,125

20. One more time