college algebra chapter 4 polynomial and rational functions

College AlgebraChapter 4

Polynomial and RationalFunctions

4.1 Polynomial Long Division and Synthetic Division

When the division cannot be completed by factoring, polynomial long division is used and closely resembles whole number division

In the division process, zero “place holders” are sometimes used to ensure that like place values will “line up” as we carry out the algorithm

32278 3 xandxofquotienttheFind

32278 3 xandxofquotienttheFind


2700832 23 xxxx

27832 3 xx


34219 32 nnnn

41923 23 nnnn


3

2142

x

xx

2143 2 xxx


If one number divides evenly into another, it must be a factor of the original number

The same idea holds for polynomials

This means that division can be used as a tool for factoring

We need to do two things first

a. Find a more efficient method for divisionb. Find divisors that give a remainder of zero


Synthetic Division

517132 23 xxxx

1 -2 -13 -175

Multiply in the diagonal direction, add in the vertical direction

1

5

3

15

2

10

-7

remainder

Explanation of why it works is on pg 376


Synthetic Division

7

73412 23

x

xxx


Synthetic Division

3

12153

x

xx


Synthetic Division and Factorable Polynomials

Principal of Factorable PolynomialsGiven a polynomial of degree n>1 with integer coefficients and a lead coefficient of 1 or -1, the linear factors of the polynomial must be of the form (x-p) where p is a factor of the constant term.

24410 234 xxxxUse synthetic division to help factor

64

83

122

241

Hint: Start with

what is easiest

1 1 -10 -4 24


Synthetic Division and Factorable Polynomials

64 23 xxx


What values of k will make x-3 a factor of 272 kxx


Homework pg 380 1-58

4.2 The Remainder and Factor Theorems

The Remainder TheoremIf a polynomial P(x) is divided by a linear factor (x-r), the remainder is identical to P(r) – the original function evaluated at r.

Use the remainder theorem to find the value of H(-5) for

6583 234 xxxxxH


Use the remainder theorem to find the value of P(1/2) for

232 23 xxxxP


The Factor Theorem

Given P(x) is a polynomial,1. If P(r) = 0, then (x-r) is a factor of P(x).2. If (x-r) is a factor of P(x), then P(r) = 0

Use the factor theorem to find a cubic polynomial with these three roots:

2,2,3 xxx


A polynomial P with integer coefficients has the zeros and degree indicated. Use the factor theorem to write the function in factored and standard form.

4 degree;1,3,7,7 xxxx


Complex numbers, coefficients, and the Remainder and Factor Theorems

Show x=2i is a zero of:

1243 23 xxxxP


Complex Conjugates Theorem

Given polynomial P(x) with real number coefficients, complex solutions will occur in conjugate pairs.

If a+bi, b≠0, is a solution, then a-bi must also be a solution.


Roots of multiplicitySome equations produce repeated roots.

Polynomial zeroes theorem

A polynomial equation of degree n has exactly n roots, (real and complex) where roots of multiplicity m are counted m times.

4.3 The Zeroes of Polynomial Functions

The Fundamental Theorem of Algebra

Every complex polynomial of degree n≥1 has at least one complex root.

Our search for a solution will not be fruitless or wasted, solutions for all

polynomials exist.The fundamental theorem combined with the factor

theorem enables to state the linear factorization theorem.


Linear factorization theorem

Every complex polynomial of degree n ≥ 1 can be written as the product of a nonzero constant and exactly n linear factors

THE IMPACT

Every polynomial equation, real or complex, has exactly n roots, counting roots of multiplicity


Find all zeroes of the complex polynomial C, given x = 1-I is a zero. Then write C in completely factored form:

ixixixxC 66521 23


The Intermediate Value Theorem (IVT)

Given f is a polynomial with real coefficients, if f(a) and f(b) have opposite signs, there is at least one value r between a and b such

that f(r)=0

HOW DOES THIS HELP???

Finding factors of polynomials


The Rational Roots Theorem (RRT)

Given a real polynomial P(x) with degree n ≥ 1 and integer coefficients, the rational roots of P (if they exist) must be of the form p/q, where p is a factor of the constant term and q

is a factor of the lead coefficient (p/q must be written in lowest terms)

List the possible rational roots for 02442143 234 xxxx


Tests for 1 and -1

1. If the sum of all coefficients is zero, x = 1 is a rood and (x-1) is a factor.

2. After changing the sign of all terms with odd degree, if the sum of the coefficients is zero, then x = -1 is a root and (x+1) is a

factor.

4.4 Graphing Polynomial Functions

THE END BEHAVIOR OF A POLYNOMIAL GRAPH

If the degree of the polynomial is odd, the ends will point in opposite directions:1. Positive lead coefficient: down on left, up on right (like y=x3)

2. Negative lead coefficient: up on left, down on right (like y=-x3)

If the degree of the polynomial is even, the ends will point in the same direction:3. Positive lead coefficient: up on left, up on right (like y=x2)

4. Negative lead coefficient: down on left, down on right (like y=-x2)


Attributes of polynomial graphs with roots of multiplicity

Zeroes of odd multiplicity will “cross through” the x-axisZeroes of even multiplicity will “bounce” off the x-axis

23 2 xxxf

bounce

Cross through


Estimate the equation based on the graph

1)³ +(x 2)² -(x = g(x)


g(x) = (x - 2)² (x + 1)³ (x - 1)²

Estimate the equation based on the graph


Guidelines for Graphing Polynomial Functions

1. Determine the end behavior of the graph2. Find the y-intercept f(0) = ?3. Find the x-intercepts using any combination

of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula.

4. Use the y-intercepts, end behavior, the multiplicity of each zero, and a few mid-interval points to sketch a smooth, continuous curve.


1. Determine the end behavior of the graph

2. Find the y-intercept f(0) = ?3. Find the x-intercepts using any

combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula.


Sketch the graph of 1249 24 xxxxg

Down, Down

F(0) = -12

3211 2 xxxxfCut through

Cut through

bounce


f(x) = x⁶ - 2 x⁵ - 4 x⁴ + 8 x³ 1. Determine the end behavior of the graph

2. Find the y-intercept f(0) = ?3. Find the x-intercepts using any

combination of the rational root theorem, factor and remainder theorems, factoring, and the quadratic formula.


4.5 Graphing Rational Functions

Vertical Asymptotes of a Rational Function

Given is a rational function in lowest

terms, vertical asymptotes will occur at the real zeroes of g

xgxf

xr

The “cross” and “bounce” concepts used for polynomial graphs can also be

applied to rational graphs 2

1

x

xf 22

1

x

xg

bouncecross


x- and y-intercepts of a rational functionGiven is in lowest terms, and x = 0 in the domain of r,

1. To find the y-intercept, substitute 0 for x and simplify. If 0 is not in the domain, the function has no y-intercept

2. To find the x-intercept(s), substitute 0 for f(x) and solve. If the equation has no real zeroes, there are no x-intercepts.

xgxf

xr

Determine the x- and y-intercepts for the function 1032

2

xx

xxh

10030

00

2

2

h 103

02

2

xx

x

20 x 00 h

0,0 0,0


Determine the x- and y-intercepts for the function 1

32

x

xh

1

30

2 x

10

30

2 h

30

No x-intercept

30 h

3,0

Y-intercept



terms, where the lead term of f is axn and the lead term of g is bxm

xgxf

xr

Polynomial f has degree n, polynomial g has degree m

1. If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis)

2. If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients)

3. If n>m, the graph of h has no horizontal asymptote

2

32

x

xxr

2

32

2

x

xxr

2

32

3

x

xxr


Guidelines for graphing rational functions pg 428

1. Find the y-intercept [evaluate r(0)]2. Locate vertical asymptotes x=h [solve g(x) = 0]3. Find the x-intercepts (if any) [solve f(x) = 0]4. Locate the horizontal asymptote y = k (check degree

of numerator and denominator)5. Determine if the graph will cross the horizontal

asymptote [solve r(x) = k from step 46. If needed, compute the value of any “mid-interval”

points needed to round-out the graph7. Draw the asymptotes, plot the intercepts and

additional points, and use intervals where r(x) changes sign to complete the graph



xgxf

xr

4.5 Graphing Rational Functions1. Find the y-intercept [evaluate r(0)]2. Locate vertical asymptotes x=h [solve g(x) =

0]3. Find the x-intercepts (if any) [solve f(x) = 0]4. Locate the horizontal asymptote y = k (check

degree of numerator and denominator)5. Determine if the graph will cross the

horizontal asymptote [solve r(x) = k from step 4

6. If needed, compute the value of any “mid-interval” points needed to round-out the graph

7. Draw the asymptotes, plot the intercepts and additional points, and use intervals where r(x) changes sign to complete the graph

7

3632

2

x

xxxr




xgxf

xr

Polynomial f has degree n, polynomial g has degree m

1. If n<m, the graph of h has a horizontal asymptote at y=0 (the x-axis)

2. If n=m, the graph of h has a horizontal asymptote at y=a/b (the ratio of lead coefficients)

3. If n>m, the graph of h has no horizontal asymptote

2

32

x

xxr

2

32

2

x

xxr

2

32

3

x

xxr

4.6 Additional Insights into Rational Functions

Oblique and nonlinear asymptotes


terms, where the degree of f is greater than the degree of g. The graph will have an oblique or nonlinear asymptote as

determined by q(x), where q(x) is the quotient of

xgxf

xr

xg

f

x

xxr

12

xx

x 12

xx

1

xxq

2

4 1

x

xxr

22

4 1

xx

x

22 1

xx

2xxq


1

42

3

x

xxxv


Choose one application problem

4.7 Polynomial and Rational Inequalities – An Analytical View

Solving Polynomial Inequalities

Given f(x) is a polynomial in standard form pg 4521. Use any combination of factoring, tests for 1 and -

1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero.

2. Plot the zeroes on a number line (x-axis) and determine if the graph crosses (odd multiplicity) or bounces (even multiplicity) at each zero. Recall that complex zeroes from irreducible quadratic factors can be ignored.

3. Use end behavior, the y-intercept, or a test point to determine the sign of the function in a given interval, then label all other intervals as P(x) < 0 or P(x) > 0 by analyzing the multiplicity of neighboring zeroes.

4. State the solution using interval notation, noting strict/non-strict inequalities.


1. Use any combination of factoring, tests for 1 and -1, the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero.




0,1834 23 xfxxxxf

Synthetic division

962 2 xxx

232 xx

bouncecross

End behavior is down/up

down

up

f(x) > 0f(x) < 0f(x) < 0

2,33, x


03523 xxx1. Use any combination of factoring, tests for 1 and -1,

the RRT and synthetic division to write P in factored form, noting the multiplicity of each zero.




Test for 1 and -1

Add coefficients 1+1+-5+3=0Means that x=1 is a root

31 2 xx

cross bounce

End behavior down/up

f(x) < 0 f(x) > 0 f(x) > 0

3,x

321 2 xxx


3

2

2

1

x

x

x

x

03

2

2

1

x

x

x

x

0

32

434 22

xx

xxx

032

74

xx

x

The graph will change signs at x = 2, -3, and

7/4

The y-intercept is 7/6 which is positive

above

belowbelow

above

,2

4

7,3x


above

below

above

Chapter 4 Review

xx

xx

2

3 42xx

xx

2

431

Chapter 4 Review

Chapter 4 Review

Use synthetic division to show that (x+7) is a factor of 2x4+13x3-6x2+9x+14

Chapter 4 Review

Factor and state roots of multiplicity

9686 234 xxxxxh

Chapter 4 Review

State an equation for the given graph

311 2 xxxxf

3242 234 xxxxxf

Chapter 4 Review

State an equation for the given graph

4

42

2

x

xxxf

Chapter 4 Review

Graph 43

92

2

xx

xxr

Chapter 4 Review

Trashketball Review

Divide using long division

xx

xx

2

822

3

xx

xx

2

8942

2

2

654 23

x

xxx

8;762 Rxx

Chapter 4 Review

Trashketball Review

Use synthetic division to divide

2

654 23

x

xxx

8;762 Rxx

7

1496132 234

x

xxxx

22 23 xxx

Chapter 4 Review

Trashketball Review

Show the indicated value is a zero of the function

1384;2

1 23 xxxxPx

Chapter 4 Review

Trashketball Review

Show the indicated value is a zero of the function

1892;3 23 xxxxPix

Chapter 4 Review

Trashketball Review

Find all the zeros of the function

Real root x=3Complex roots x=±2i

Chapter 4 Review

Trashketball Review

Find all the zeros of the function

Chapter 4 Review

Trashketball Review

State end behavior, y-intercept, and list the possible rational roots for each function

Chapter 4 Review

Trashketball Review

Sketch the Graph using the degree, end behavior, x- y-intercept, zeroes of multiplicity and midinterval points

Chapter 4 Review

Trashketball Review

Graph using guidelines for graphing rational functions

college algebra chapter 4 polynomial and rational functions

Documents

polynomial long division

factor of slide

number division

division process

division b

polynomial px

synthetic division homework

factor of px