1. Finding Zeros of a Polynomial Function

2. Fundamental Theorem of Algebra (FTA)
every polynomial has at least one solution

3. Fundamental Theorem of Algebra (FTA) COROLLARY
The degree (biggest exponent) = # of roots

4. These words mean the same thing
Roots
Zeros
Solutions
X intercepts
A factor is just x - #

5. Decarte’s Rule of Signs
The number of times the sign changes in p(x)= possible number of positive roots
Or 2 less, 4 less, 6 less, etc

6. Decarte’s Rule of Signs
The number of times the sign changes in p(-x)= possible number of negative roots
Or 2 less, 4 less, 6 less, etc

7. Decarte’s Rule of Signs
Can sometimes narrow down which numbers to check
Can also tell how many imaginary roots are possible.
Degree – (# of + plus # -)

8. Ex with Decarte’s Rule CHECK NEGATIVE NUMBERS 4x^3 – 7x + 3 = p(x)
Signs in order [p(x)] + - +
There are 2 or 0 positive roots
P(-x)= 4(-x)^3-7(-x)+3
=-4x^3 + 7x + 3
Signs in order for p(-x): - + +
There is only 1 sign change
We are guaranteed 1 negative root
CHECK NEGATIVE NUMBERS

9. Rational Zero Theorem
P = all the numbers you can multiply to get the constant
Q = all the numbers you can multiply to get the leading coefficient
+- p/q = all POSSIBLE factors of your polynomial

10. Upper bound/lower bound
Will cover these 2 in 4-5
Tells us there will be no roots above #
Tells us there will be no roots below #
Uses synthetic division

11. Location Principal Helps find fractional and irrational zeros
Uses synthetic division or graph

12. Factor Theorem When using synthetic division, if the remainder is 0, then the # you divided by is a root, zero, solution, x-intercept AND X – divisor Is a factor!

13. Put it together…

14. Finding zeros Location Principal Factor Thm Upper/Lower bounds thms
Decarte’s Rule
Rational Zero thm
FTA and it’s corollary

15. Decarte’s Rule of Signs
Organizational Chart
FTA
Rational Zero Thm
Decarte’s Rule of Signs
Upper/Lower Bound Thm
Factor Thm
Location Principal
Solve Quadratics

16. Let’s do an example 4x^3 – 7x +3 FTA– there is at least one root
Corollary
There are 3 roots
Decarte’s rule
2 or 0 positive
1 negative
2 or 0 imaginary

17. 4x^3 – 7x +3 continued Rational Zero Thm + or -, 1,3,1/2,1/4,3/2,3/4
We know that we are guaranteed 1 negative root
Start checking negative roots
While checking, notice the quotient– if all positive #’s, that’s the upper bound
While checking, notice the quotient– if signs alternate, that’s a lower bound

18. 4x^3 – 7x +3 continued While checking, notice the remainder
What does the factor thm say?
When we divide by – 3/2 the remainder is zero:
-3/

19. 4x^3 – 7x +3 continued Quotient: 4x^2 -6x + 2 Factors as
While checking consecutive integers, check for sign change in the remainder location principal
We have all 3 zeros
1, ½, -3/2