Honors Pre-Calc UNIT 5 POLYNOMIALS REVIEW General form of a polynomial: _____________________________________________ Types of Polynomials: Linear, Quadratics, Cubics, Quartics, … Polynomials are “smooth and continuous” functions. If degree is even: If degree is odd: HIGHER ORDER POLYNOMIALS: Rules:

§ To find ALL the zeros of the function, use one real root found in the calculator to do synthetic division, then solve the remaining quadratic.

§ Remainder Theorem: f(k) = the remainder when dividing a polynomial by (x – k). § Factor Theorem: when f(k) = 0, x – k is a factor of f(x). § When a zero is repeated, it has a multiplicity of 2. If a root occurs 3 times, multiplicity of 3… § Fundmental Theorem of Algebra: The degree (n)= number of roots. n – 1 = maximum number of

extrema. Ex: Find the remainder when you divide x3 – 6x + 4 by x – 2. Is x – 2 a factor? Ex 2: Find the remainder when you divide x27 – 1 by x + 1. Is x + 1 a factor? Ex 3: f(x) = (x – 3)2(x + 5)3 Find the roots and state the multiplicity of each: What is the degree? ___ # of zeros? ____ # of possible extrema? ____ Actual # of extrema? ____ End Behavior? Ex 4: Find the roots and state the multiplicity of each: What is the degree? ___ # of zeros? ____ # of possible extrema? ____ Actual # of extrema? ____ End Behavior? Write the linear factorization:

03523 =--- xxx

Aireyf x AnXnt An x t t A X ta

Pos Coeff A Pos Coeff Neg Coeff Inofcxs oo IhmIff a fight is Infested

hi f fCxO YI fCx P

f 27 273 612 4 0 remainder f tibscarohWff D f 1 I I 1 2 remainders Tarqagtory

L5 11 3 multiplicity of 254 X 5 multiplicity of 3NONE

YsfCx oo hj7 fCx as

3 3 I I 5 3 11 332 k

3 I multiplicity2

3 fades hj7fCxas

of 2

X 3 Xt 120

Ex 5: ! = #$ − 4#' + 9# − 36 Find the roots and state the multiplicity of each: What is the degree? ___ # of zeros? ____ # of possible extrema? ____ Actual # of extrema? ____ End behavior? Write the linear factorization: Ex 6: ,(#) = #/ + 5#$ + 7#' + 5# + 6 Find the roots and state the multiplicity of each: What is the degree? ___ # of zeros? ____ # of possible extrema? ____ Actual # of extrema? ____ End behavior? Write the linear factorization: ___ 7. In the equation , 2 is a root with multiplicity ___.

a) 0 b) 2 c) 3 d) 1 e) 4

___ 8. Looking at the synthetic division shown, what is the complete factorization of

f(x) = ?

a) (x-2)(x-2)(x+1)(x-3)

b) (x-2)(x+1)(x-3)

c) (x+2)(x+2)(x-1)(x+3)

d) (x+2)(x-1)(x-3)

___ 9. One root of is 2i. Find the others.

a) 4i, 0 b) -2i, 1 c) -2i, -1 d) 0, -1 e) 2i, -2i

0242863 234 =-+-- xxxx

12496 234 -++- xxxx

04423 =-+- xxx

2 1 -6 9 4 -12

2 -8 2 12

-1 1 -4 1 6 0

-1 5 -6

3 1 -5 6 0

3 -6

1 -2 0

3 4J I 4 9 3632 440 36NONE O 9 WO

zfc A KI fCx as 1179 02 9 3i

X 4 x 3i xt3i

44 3J 15 7 56

3 icyerY Hx O XIII 46 2 I 3 628

cxt3Xxt2 Cx i CXti z

t0YA

X 2 X 2

XH 11 1

pine

q X 2BX 0 X X X

I I 4 4x t.tt o4aiiw

i

___ 10. Find the roots of .

a) -1,4,9 b) -3,7,9 c) -1,2,4 d) -3,6,7 e) 1,-4,-9

Find ALL roots of the polynomial given. Leave ALL answers exact (NO DECIMALS!). Show work!

11. g(x) = 12. h(x) =

13. Write the equation of a fourth degree polynomial in expanded form with roots 3, -2, and -3 + i.

14. Use synthetic division to find the value of k so that the remainder for is 10.

Simplify.

15. 16. 17. 18.

19. Given f(x) = (x + 6)2(x – 4)5.

a. State the roots and their multiplicity.

b. What is the degree?

c. A polynomial with the same degree as f(x) could have ____ extrema,

d. Without graphing, state the end behavior of f(x). Use limits!

e. Sketch the graph without a calculator.

20. Solve the following inequalities:

a) b)

c) d)

0362312 23 =++- xxx

123219 234 -+-+ xxxx 5444 234 --++ xxxx

( ) ( )125 23 -÷++- xkxxx

14i ( )( )ii 4273 +-ii2325

+- ( )( )2441812 -+--+

A0 X X

4 F4U I l 19 32 12 I I 4 4 4 5 I

oI IIX 31 2 Xt3 i Xt3ti x'tax to

Tx E Et tI l 5 2 K k

datzira f4t4ir23 2i3 ai

644 15 10 hfg f a aeX 6 m of Z 11 4 mot 5

76YY f x 0 415 of

6,3 too 2 VfB 3 U 6,0

8,0 8 77012,0

Cumulative Review:

21. Find the inverse of .

22. The number of North Carolina cows infected with the mad cow disease after t days is modeled by the function 3(4) = 56789/:;?@. a) What is the initial cows that were infected?

b) What are the horizontal asymptotes?

c) State the end behavior

d) When will it reach 473 cow infected?

23. The following table depicts the spending at the National Institute of Health (NIH). Let 1990 be year 0.

a. Find the linear regression: b. Find the quadratic regression: c. Use both of your models to estimate when the spending will exceed 30 billion.

Linear: Quadratic:

SOLVE & SHOW ALL NECESSARY WORK STATE EXTRANEOUS VALUES (these could be non-calc btdubs)

24. 25. log' 8$' = #

26. 27.

28. Describe the patterns, find the next 3 terms, write the recursive rule, and write the explicit formula. 13, 15, 17, 19, ... 4, 8, 16, 32, ...

5223)(

+-

=xxxf

3log4 =x

( ) 1log9log =+- xx ( ) ( )234 3log615loglog xxxx --=

Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002

$ (billions)

10.3 11.0 11.3 11.9 12.7 13.6 15.6 17.9 20.5 23.6

57845

12.84

4 0 9 578high.gg i 578YhfcsPlt Oe ut3eoo4826473Cit46e43tJ szg en en4739,74687se 46eo43t zygq E 73g33846e43t zziaa t l246 To3 45 6 89 te

4 1.40 1 4.33YI 1875 2 1.411 13.33

t 19 E 14

2 425 1

434 2 Yz 2 2 5640 211 32 1150

1129 10 0429171 ex toCxtD o log log

534763 5 3 0Y x 5 2 2x 5 310 X2_9 74747 5 2 3 05 3 0

An_Ant 2Apping2eachtimetsArithmetic 0242325 An 2 n 1 13 _2h 2 13 2h 111

Multiplyby2eachtimetGeometric 064,128,256 An Any 2An 4 27