Chapter 3- Polynomial and Rational Functions

16 Days

3.1 Polynomial Functions of Degree

Greater Than 2One Day

How does a mountain range relate to polynomials and their graphs?

Essential Question

Definition: If f is a polynomial function and f(a) ≠f(b) for a<b, then f takes on every value between f(a) and f(b) in the interval [a, b].

Facts about IVT: f is continuous from f(a) to f(b) If f(a) and f(b) have opposite signs (1

positive and 1 negative), there is at least one number c between a and b such that f(c)=0.

Intermediate Value Theorem

Diagram of “IVT”

Step 1◦ Identify a and b values

Step 2◦ Plug a and b into f(x) and solve

Step 3◦ If f(a) and f(b) have opposite signs, you know that

f(c)=0 for at least one real number between a & b

Steps Using IVT

Using the Intermediate Value Theorem Show that f(x)= x⁵+2x⁴-6xᶟ+2x-3 has a zero

between 1 and 2.◦ Step 1: identify a and b

a=1, b=2◦ Step 2: solve f(a) and f(b)

f(a)= f(1)= 1+2-6+2-3=-4 f(b)=f(2)=32+32-48+4-3=17

◦ Step 3: do f(a) and f(b) have opposite signs? f(a)=-4 and f(b)=17, yes!

◦ Conclusion: there is a c where f(c)=0 between 1 and 2

Example of IVT

Sketching graphs of degree greater than 2◦ Step 1

Find the zeros◦ Step 2

Create a table showing intervals of positive or negative signs for f(x).

◦ Step 3 Find where f(x)>0 and where f(x)<0

Sketching Graphs

Construct a table showing intervals of positive or negative values of f(x)

Step 2 Explained

Interval

Sign of (x-1) - + +

Sign of (x-4) - - +

Sign of f(x) + - +

Position of graph

Above x-axis

Below x-axis

Above x-axis

1, ),4( )4,1(

•This means that f(x)>0 if x is in •This means that f(x)<0 if x is in .

),4(1, and)4,1(

Sketching the graph of a polynomial function of degree 3◦ Ex

Step 1: Find Zeros

◦ Group terms

◦ Factor out x² and -4

◦ Factor out (x+1)

◦ Difference of squares

◦ Therefore, the zeros are -2, 2, and -1

Sketching Graphs

44)( 23 xxxxf

)44()()( 23 xxxxf

)1(4)1()( 2 xxxxf

)1)(4()( 2 xxxf

)1)(2)(2()( xxxxf

Step 2: Create a Table

Sketching Graphs

Interval

Sign of x+2

Sign of x+1

Sign of x-2

Sign of f(x)

Position of graph

)2,( )1,2( )2,1( ),2(

f(x)>0 if x is in f(x)<0 if x is in My version of graph (there are multiple

ways to do this graph)

Sketch a Graph),2()1,2(

)2,1()2,(

x

y

Ex Find the intervals

◦ f(x) is below x-axis when x is in ◦ f(x) is above x-axis when x is in

Sketch graph

Sketching a graph given sign diagram

)2,0()3,( and),2(),0,1(),1,3( and

x

y

Steps◦ Step 1: Assign f(x) to Y1 on a graphing calculator◦ Step 2: Set x and y bounds large enough to see

from [-15,-15] and [-15,15] This allows us to gauge where the zeros lie from a

broad perspective◦ Step 3: Readjust bounds once you know where

zeros are more likely to be found◦ Step 4: Use zero or root feature on calculator to

estimate the real zero

Estimating Zeros

Ex/ Estimate the real zeros of

Graph this function as Y1 in calculator Set bounds as [-15,15] by [-15,15]. Readjust to where zeros might exist, you may use [-1,3] by [-1,3] Find actual root by using zero

or foot feature on your calculator

Estimating Zeros

656.072.56.4)( 23 xxxxf

Actual root is 0.127

P. 227 #1,5,7,15,16,18,23,25,34,41,50

If you need help, check with the person beside you.

Homework

3.2 Properties of Division

Two Days

We will first divide 2172 by 3 to find the quotient and remainder.

Let’s review long division…

If g(x) is a factor of f(x), then f(x) is divisible by g(x).

For example, is divisible by each of the following .

Notice that is not divisible by .

However, we can use Polynomial Long Division to find a quotient and a remainder.

Properties of Division

814 x3 and ,3 ,9 ,9 22 xxxx

814 x 132 xx

Step 1: Make sure the polynomial is written in descending order. If any terms are missing, use a zero to fill in the missing term (this will help with the spacing).

Step 2: Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol.

Step 3: Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol.

Step 4: Subtract and bring down the next term.

Step 5: Repeat Steps 2, 3, and 4 until there are no more terms to bring down.

Step 6: Write the final answer. The term remaining after the last subtract step is the remainder and must be written as a fraction in the final answer.

http://www.youtube.com/watch?v=smsKMWf8ZCs

Steps for dividing polynomials

Divide by .

Lets do another..9272 23 xxx 32 x

927232 23 xxxx22

15

64

94

64

24

32

927232

2

2

2

23

23

xx

x

x

xx

xx

xx

xxxx

32

1522

32

9272 :Solution 2

23

xxx

x

xxx

Divide by .

One More Practice Problem..

1243 23 xxx 62 xx

12436 232 xxxxx2

0

1222

1222

6

12436

2

2

23

232

x

xx

xx

xxx

xxxxx

26

1243 :Solution

2

23

x

xx

xxx

If a polynomial f(x) is divided by x-c, the the remainder is f(c).

Example:

Remainder Theorem

f(-2). find totheoremremainder theuse 5,3)( If 23 xxxxf

532 23 xxxx

-17.(-2) Therefore, f

A polynomial f(x) has a factor x-c if and only if f(c)=0.

Example:

Factor Theorem

2.34)( offactor a is 2- that xShow 23 xxxxf

Find a polynomial f(x) of degree 3 that has zeros 3, -1, and 1.

Finding a Polynomial with Known Zeros

pg 236 (# 1,2,8,17-20)

Homework

http://www.mesacc.edu/~scotz47781/mat120/notes/divide_poly/long_division/long_division.html

For Extra Examples

3.2 Day 2 – Synthetic Division

Synthetic Division is the “shortcut” method for dividing polynomials.

Synthetic Division

Long vs. SyntheticDivision of Polynomials

division.synthetic and division

long both using 1by 22 dividing compare sLet' 34 xxxx

422

6

44

24

22

22

22

02

2201

23

2

2

23

23

34

234

xxx

x

x

xx

xx

xx

xx

xx

xxxxx

Long Division Synthetic Division

422

64221

4221

220111

23

xxx

Use synthetic division to find the quotient q(x) and remainder r

Lets try a few

.3by divided is 5,34)( if 234 xxxxxf

Using synthetic division to find zeros.◦ What must we show for a value to be a zero of

f(x)? Think Factor Theorem…

More Synthetic Division

44.298)( of zero a is 11- that Show 23 xxxxf

Use synthetic division to find f(3) if

More Synthetic Division

1533)( 234 xxxxxf

You should now recognize the following equivalent statements. If f(a)=b, then:◦ 1. The point (a,b) is on the graph of f.◦ 2. The value of f at x=a equals b; ie f(a)=b.◦ 3. If f(x) is divided by x-a, then the remainder is b.

Additionally, if b=0 then the following are also equivalent. ◦ 1. The number a is a zero of the function f. ◦ 2. The point (a,0) is on the graph of f; a is an x-int.◦ 3. The number a is a solution of the equation

f(x)=0.◦ 4. The binomial x-a is a factor of the polynomial

f(x).

Check Point

pg 237 (# 21-29 odd,32,35,38)

Homework

3.3 Zeros of Polynomials

Two Days

If a polynomial f(x) has positive degree and complex coefficients, then f(x) has at least one complex zero.

Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n>0, then there exist n complex numbers such that

Where a is the leading coefficient of f(x). Each number is a zero of f(x).

Complete Factorization Theorem for Polynomials

nccc ,...,, 21

),)...()(()( 21 ncxcxcxaxf

kc

A polynomial of degree n>0 has at most n different complex zeros.

Theorem on the Maximum Number of Zeros of a Polynomial

Multiplicities of Zeros

f f

f

End Behavior of Polynomials

Find the zeros of the polynomial, state the multiplicity of each zero, find the y-int, and sketch the graph.

Finding Multiplicities

23 )1()3)(2(25

1)( xxxxf

Find a polynomial f(x) in factored for that has degree 3; has zeros 3, 1, and -1; and satisfies f(-2)=3.

Finding a Polynomial with Prescribed Zeros

If f(x) is a polynomial of degree n>0 and if a zero of multiplicity m is counted m times, then f(x) has precisely n zeros.

Exactly how many zeros exist for the following polynomial?

Theorem on the Exact Number of Zeros of a Polynomial (N-zeros Theorem)

523 )2()1()3)(2(3)( xxxxxf

pg 249 (# 1-7 odd, 11-15 odd, 19-23 odd)

Homework

If f(x) is a polynomial with real coefficients and a nonzero constant term, then:

◦ 1. The number of positive real zeros of f(x) either is equal to the number of variations of sign in f(x) or is less than that number by an even integer.

◦ 2. The number of negative real zeros of f(x) either is equal to the number of variations of sign in f(-x) or is less than that number by an even integer.

Descartes’ Rule of Signs

We can use the N-Zeros Thm to determine the total number of zeros possible.

Use Descartes’ Rule of Signs to determine the total possible number of positive and negative zeros (and all lesser combinations).

Any unaccounted for zeros must then be imaginary zeros.

Using Descartes’ Rule of Signs

Find the total number of zeros possible for f(x)=0 where

Example of Descartes’

56323)( 345 xxxxxf

Number of positive solutions

Number of negative solutions

Number of imaginary solutions

Total number of solutions

Suppose that f(x) is a polynomial with real coefficients and a positive leading coefficient and that f(x) is divided synthetically by x-c. Then:◦ 1. If c>0 and if all numbers in the third row of the

division process are either positive or zero, the c is a upper bound for the real zeros of f(x).

◦ 2. If c<0 and if the numbers in the third row of the division process are alternately positive and negative (a 0 can be counted as positive OR negative), then c is a lower bound for the real zeros of f(x).

First Theorem on Bounds for Real Zeros of Polynomials

Determine the smallest and largest integers that are upper and lower bounds for the real solutions of f(x)=0 if

Example of 1st Thm on Bounds

632)( 234 xxxxxf

Suppose is a polynomial with real coefficients. All of the real zeros of f(x) are in the interval

Second Theorem on Bounds for Real Zeros of Polynomials

011

1 ...)( axaxaxaxf nn

nn

1

,,...,,max e wher,),( 011

n

nn

a

aaaaMMM

Determine the interval in which all of the real solutions of f(x)=0 exist if

Example of 2nd Thm on Bounds

56323)( 345 xxxxxf

Finding Zeros Practice WS (# 1-9) Quiz Tomorrow

Homework

3.4 Complex and Rational Zeros of Polynomials

Two Days

If a polynomial f(x) of degree n>1 has real coefficients and if with b≠0 is a complex zero of f(x), then the conjugate . is also a zero of f(x).

Theorem on Conjugate Pair Zeros of a Polynomial

biaz

biaz

Find a polynomial of degree 3 with real coefficients where 2 and (3-i) are zeros of the polynomial.

Example

Every polynomial with real coefficients and positive degree n can be expressed as a product of linear and quadratic factors with real coefficients such that the quadratic factors are irreducible over ℝ.

Theorem on Expressing a Polynomial as a Product of Linear and Quadratic Factors

If the polynomial

has integer coefficients and if is a rational zero of f(x) such that c and d have no common prime factor, then:

1. the numerator, c, of the zero is a factor of the constant term

2. the denominator d of the zero is a factor of the leading coefficient

Rational Zeros Theorem

011

1 ...)( axaxaxaxf nn

nn

d

c

All possible rational zeros of a polynomial are of the form:

Using Rational Zeros Theorem

na

a

t coefficien leading theof factors

ermconstant t theof factors ZerosRational Possible 0

Find all possible rational zeros using rational zeros theorem. Then factor and find any remaining zeros.

Find all zeros of f(x)

41223)( 234 xxxxxf

pg 259 (# 7,18,19,23,24,29) Top Shelf pg 14 (#1,2,5,6,8; Due 11/11)

Homework

Prentice Hall p 46 Fundamental Thm Algebra WS (# 1-4,8,9,10,13,17)

Homework

3.5 Rational Functions

Four Days

A function f is a rational function if

where g(x) and h(x) are polynomials.

The domain of f consists of all real numbers except the zeros of the denominator h(x). A zero of h(x) that reduces results in a hole in the graph instead of a vertical asymptote.

Rational Functions

)(

)()(

xh

xgxf

Find the domain of the following rational functions:

Rational Functions and Domain

32

1)(

xxf

23

14)(

2

xx

xxg

9

27)(

2

3

x

xxg

,, :D 23

23 ,11,22, :D :D

Simplify and graph the following rational expression:

Simplifying Rational Expressions

2

65)(

2

x

xxxf

Definition: The line x=a is a vertical asymptote for the graph of the function f if

as x approaches a from either the left of the right.

Vertical Asymptotes

axaxxfxf or as )(or )(

Create an xy-table and look at the values of f(x) as x approaches the vertical asymptote.

What is really happening?

1

1)(

xxf

Definition: The line y=c is a horizontal asymptote for the graph of the function f if

Horizontal Asymptotes

xxcxf or as )(

Theorem on Horizontal Asymptotes

asymptote. horizontal no is there, When 3.

asymptote. horizontal the

is )coeffients leading of (ratio line the, When 2.

asymptote. horizontal theis 0 axis- x the, When 1.

factors.common no with spolynomial are and where

bygiven function rational a be Let

011

1

011

1

mn

b

a y mn

ymn

xdxn

bxbxbxb

axaxaxa

xd

xnxf

f

m

n

mm

mm

nn

nn

If a factor can be reduced that would cause a vertical asymptote there will be a hole instead of a vertical asymptote!

Holes and Asymptotes

45

1)(

2

2

xx

xxf

Find the Vertical and Horizontal Asymptotes

105

14)(

x

xxf

Find the Vertical and Horizontal Asymptotes

9

13)(

2

2

x

xxf

Find the Vertical and Horizontal Asymptotes

65

1)(

2

xx

xxf

Find the Vertical and Horizontal Asymptotes

65

1)(

2

xx

xxf

pg 276 (# 1,2,3,5,6,8,11,15)◦ Only find the Domain, VA, HA, and reduced form

of the rational function. ◦ We will graph them tomorrow!

Homework

Sketching Rational Functions

below.or above fromHA theapproachesgraph he whether tdecide to5 guideline Use

HA. or the axis- x thebelowor above isgraph he whether t tell tos value

function specific ofsign theuse necessary, If 2. guidelinein VAs by the

determined plane- xyin the regions theofeach in ofgraph Sketch the 6.

exist. they if points, Plot these .equation the to

solutions theareon intersecti of points theof scoordinate- xThe

graph. theintersectsit whether determine ,HA a is thereIf 5.

line. dashed ait with sketch , asymptote

horizontal a is thereIf .asymptotes horizontalon theoremApply the 4.

.00point plot the and exists,it if ,0intercept -y theFind 3.

line. dashed a with asymptote verticalsketch the , zero

realeach For .r denominato theof zeros real theFind 2.

axis.- xon the points ingcorrespond Plot the

. of zeros real all i.e. ,intercepts- x theFind 1.

factors.common no with spolynomial are and where

that Assume01

11

011

1

f

cf(x)

cy

cy

)),f(()f(

axa

h(x)

g(x)

xhxg

bxbxbxb

axaxaxa

xh

xgxf

mm

mm

nn

nn

Graphing Rational Functions

32

14)(

x

xxf

Graphing Rational Functions

32

1)(

2

xx

xxf

Graphing Rational Functions

43

25)(

2

2

xx

xxf

pg 46 Dennison PreCalc (find VA, HA, SA, y-int, graph # 5, 8) pg 276(# 21, 25)

Homework

Oblique Asymptotes

pg 48 Dennison PreCalc (find VA, HA, SA, y-int) pg 277 (# 29,31)

Homework

pg 276 (# 7,12,32,35,41,42) Quiz on Graphs Tomorrow

◦ (No Graphing Calculators!)

Tomorrow nights homework: pg 279 (# 1,4,9,10,11,13,16-19,24,25,27,

29,31,35)

Homework