1. 5.4 Finding All Zeros of a Polynomial

2. There are 3! Ex 1) Find all zeros of 1 1 -1 2 Zeros: -1 1 -1 3 -2 1 -1
Poss. Rational Zeros:
Since we know there’s one neg. real, try neg. first
Descartes:
(+): 2 or 0
(–):
: 1
Zeros: -1 1 -1 3 -2 1 -1 1

3. Ex 2) Find all solutions to
There are 4!
Ex 2) Find all solutions to
Poss. Rational Zeros:
Descartes:
(+): 1
(–):
: 3 or 1
Since we know there’s one pos. real, try pos. first 1 1 2 -6
-10 2 1 3 4 2 -1 1 2 2

4. Ex 2 cont… 2 is our (+) real root –1 is a (–) real root
And we know is a factor
and solve for x
~ use quadratic formula
Zeros:

5. Example 3 Find all rational zeros of 1 1 2 4 -8 1 1 2 4 8 -1 1 2 2 -10
Poss. Rational Zeros:
Descartes:
(+): 1
(–):
: 3 or 1 1 1 2 4 8
There’s only 1 pos real -1 1 2 2
-10 -2 1 -1 4 -4
Lower Bound
Rat. Zeros:
1, –2

6. T.O.O. Example 4 Find all rational zeros of 1 0 1 1 -2 1 1 1 2 3 1 -1
Poss. Rational Zeros:
Descartes:
(+): 1
(–):
: 1
I.V.T. 1 1 1 2 3 1
Upper Bound -1 1 -1 2 -1 -1
I.V.T. -2 1 -2 5 -9 16
Lower Bound
No Rational Zeros

7. Homework
#508 Pg odd, 33, 37, 41