1. 4.3: Real Zeroes of Polynomials Functions
November 14, 2008

2. P #9-18, 36-38,67-73 9)
a. 2 turning points, x-intercept = -3
b. Positive
c. Minimum degree of 3
10)
4 turning points, x-intercepts = -2, -1, 0, 1, 2
Negative
Minimum degree is 5

3. 11-12 11) 12) One turning point, x-intercepts = -1, 2 Positive 2
No turning points, x-intercept = ½
Negative 1

4. 13-15
13) (a) d (b) (1,0) (c) x=1 (d) (1,0) (e) (1,0) 14) (a) c (b) (-1, -2), (1,2) (c) x= -1.8, 0, 1.8 (d) (-1, -2), (1,2) (e) none 15) (a) b (b) (-3, 27), (1,-5) (c) x=0, 1.9 (d) (-3, 27), (1,-5) (e) none

5. 16-18
16) (a) f (b) (-2, -16), (0, 0), (2, 16) (c) x= -2.8, 0, 2.8 (d) (-2, -16), (0, 0), (2, 16) (e) (-2, -16), (2, 16) 17) (a) a (b) (-2, 16), (0,0), (2, 16) (c) x= -2.8, 0, 2.8 (d) (-2, 16), (0,0), (2, 16) (e) (-2, 16), (2, 16) 18) (a) e (b) (-2, 1), (-1, -2), (0,0), (1, -3) (c) x= -2.2, 0, 1.2 (d) (-2, 1), (-1, -2), (0,0), (1, -3) (e) none

6. 36-38
36) (b) (-2, -5.33…), (2, ) (c) both are local extrema 37)(b) (-2, ), (1, …) (c) Both are local extrema 38) (b) (-2, ), (1, 2.1), (1, -.58) (c)All are local extrema

7. 67-73
67) f(-2)=5, f(1)=0 68) f(-1)=0, f(0)= -.7, f(3)=2 69) f(-1)=-1, f(1) =1, f(2)= -2 70) f(-2)=0, f(0)=-3, f(2)=2 71) f(-3)=-63, f(1)=3, f(4)=10 72) f(-4)=16, f(0)=2, f(4)=-12 73) f(-2)=6, f(1)=7, f(2)=9

8. Objectives Divide polynomials
Understand the division algorithm, remainder theorem, and factor theorem
Factor higher degree polynomials
Analyze polynomials with multiple zeroes
Find rational zeros
Solve polynomial equations

9. Divide polynomials
Divide by a monomial
(3x3+6x2+7)/(2x)

10. Divide polynomials
Dividing by binomials or higher
(2x3+4x2-x+6)/(x+1)

12. Understand the division algorithm, remainder theorem, and factor theorem
f(x) = d(x) * q(x) + r(x)
Remainder theorem
if f(x) is divided by x-k, the remainder is f(k)
Factor theorem
f(x) has x-k as a factor if and only if f(k)=0

14. Things to remember If f(x) has a degree of 1 or greater, then…
The graph of y=f(x) has a x-intercept k.
A zero of f(x) is k. Basically, f(k)=0.
A factor of f(x) is (x-k).

15. Multiplicity
Even multiplicities (squared, to the fourth, etc.) intersect but do not cross at zero.
Odd multiplicities (1st, 3rd, etc.) cross at zero.
Total multiplicity must add up to the degree.

17. . If r is a Zero of Even Multiplicity
If r is a Zero of Odd Multiplicity .

18. Find zeros on this graph.

19. Check this out
What makes f(x) equal zero on this graph.

20. Factoring…
2x3-4x2-10x+12, given that k=2 and is a zero.

21. Your assignment
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31-38
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63-65