Polynomial and Rational Functions Chapter 3 TexPoint fonts used in EMF. Read the TexPoint manual...

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Polynomial and Rational Functions Chapter 3

Transcript of Polynomial and Rational Functions Chapter 3 TexPoint fonts used in EMF. Read the TexPoint manual...

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Polynomial and Rational Functions

Chapter 3

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Quadratic Functions and Models

Section 3.1

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Quadratic Functions

Quadratic function: Function of the form

f(x) = ax2 + bx + c(a, b and c real numbers, a ≠ 0)

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Quadratic Functions

Example. Plot the graphs of

f(x) = x2, g(x) = 3x2 and

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Quadratic Functions

Example. Plot the graphs of

f(x) = {x2, g(x) = {3x2 and

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Parabolas

Parabola: The graph of a quadratic functionIf a > 0, the parabola opens upIf a < 0, the parabola opens

downVertex: highest / lowest point

of a parabola

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Parabolas

Axis of symmetry: Vertical line passing through the vertex

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Parabolas

Example. For the function f(x) = {3x2 +12x { 11

(a) Problem: Graph the function Answer:

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Parabolas

Example. (cont.) (b) Problem: Find the vertex and

axis of symmetry. Answer:

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Parabolas

Locations of vertex and axis of

symmetry:Set

Set

Vertex is at:

Axis of symmetry runs through

vertex

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Parabolas

Example. For the parabola defined by

f(x) = 2x2 { 3x + 2

(a) Problem: Without graphing,

locate the vertex.

Answer:

(b) Problem: Does the parabola

open up or down?

Answer:

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x-intercepts of a Parabola

For a quadratic function f(x) = ax2 + bx + c: Discriminant is b2 { 4ac.Number of x-intercepts depends on

the discriminant.Positive discriminant: Two x-interceptsNegative discriminant: Zero x-

interceptsZero discriminant: One x-intercept

(Vertex lies on x-axis)

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x-intercepts of a Parabola

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Graphing Quadratic Functions

Example. For the function f(x) = 2x2 + 8x + 4

(a) Problem: Find the vertex Answer:

(b) Problem: Find the intercepts. Answer:

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Graphing Quadratic Functions

Example. (cont.)(c) Problem: Graph the function

Answer:

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Graphing Quadratic Functions

Example. (cont.)(d) Problem: Determine the

domain and range of f. Answer:

(e) Problem: Determine where f is increasing and decreasing.

Answer:

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Graphing Quadratic Functions

Example. Problem: Determine the

quadratic function whose vertex is (2, 3) and whose y-intercept is 11.

Answer:

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Graphing Quadratic Functions

Method 1 for Graphing Complete the square in x to

write the quadratic function in the form y = a(x { h)2 + k

Graph the function using transformations

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Graphing Quadratic Functions

Method 2 for Graphing Determine the vertex Determine the axis of

symmetry

Determine the y-intercept f(0) Find the discriminant b2 { 4ac.

If b2 { 4ac > 0, two x-intercepts If b2 { 4ac = 0, one x-intercept

(at the vertex) If b2 { 4ac < 0, no x-intercepts.

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Graphing Quadratic Functions

Method 2 for Graphing Find an additional point

Use the y-intercept and axis of symmetry.

Plot the points and draw the graph

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Graphing Quadratic Functions

Example. For the quadratic function

f(x) = 3x2 { 12x + 7 (a) Problem: Determine whether

f has a maximum or minimum value, then find it.

Answer:

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Graphing Quadratic Functions

Example. (cont.)(b) Problem: Graph f

Answer:

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Quadratic Relations

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Quadratic Relations

Example. An engineer collects the following data showing the speed s of a Ford Taurus and its average miles per gallon, M.

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Quadratic Relations

Speed, s Miles per Gallon, M

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35 20

40 23

40 25

45 25

50 28

55 30

60 29

65 26

65 25

70 25

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Quadratic Relations

Example. (cont.)(a) Problem: Draw a scatter

diagram of the data Answer:

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Quadratic Relations

Example. (cont.)(b) Problem: Find the quadratic

function of best fit to these data.

Answer:

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Quadratic Relations

Example. (cont.)(c) Problem: Use the function to

determine the speed that maximizes miles per gallon.

Answer:

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Key Points

Quadratic FunctionsParabolasx-intercepts of a ParabolaGraphing Quadratic

FunctionsQuadratic Relations

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Polynomial Functions and Models

Section 3.2

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Polynomial Functions

Polynomial function: Function of the form

f(x) = anxn + an {1xn {1 + + a1x + a0

an, an {1, …, a1, a0 real numbers

n is a nonnegative integer (an 0)

Domain is the set of all real numbers Terminology

Leading coefficient: an

Degree: n (largest power) Constant term: a0

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Polynomial FunctionsDegrees:

Zero function: undefined degreeConstant functions: degree 0.(Non-constant) linear functions:

degree 1.Quadratic functions: degree 2.

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Polynomial Functions Example. Determine which of the

following are polynomial functions? For those that are, find the degree.

(a) Problem: f(x) = 3x + 6x2

Answer: (b) Problem: g(x) = 13x3 + 5 + 9x4

Answer: (c) Problem: h(x) = 14

Answer: (d) Problem:

Answer:

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Polynomial Functions

Graph of a polynomial function will be smooth and continuous. Smooth: no sharp corners or cusps. Continuous: no gaps or holes.

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Power Functions

Power function of degree n:Function of the form

f(x) = axn

a 0 a real numbern > 0 is an integer.

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Power Functions

The graph depends on whether n is even or odd.

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Power Functions

Properties of f(x) = axn

Symmetry:If n is even, f is even.

If n is odd, f is odd.

Domain: All real numbers.

Range:If n is even, All nonnegative real

numbers

If n is odd, All real numbers.

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Power Functions

Properties of f(x) = axn

Points on graph:If n is even: (0, 0), (1, 1) and ({1, 1)

If n is odd: (0, 0), (1, 1) and ({1, {1)

Shape: As n increases Graph becomes more vertical if |x| > 1

More horizontal near origin

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Graphing Using Transformations

Example. Problem: Graph f(x) = (x { 1)4 Answer:

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Graphing Using Transformations

Example. Problem: Graph f(x) = x5 + 2Answer:

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Zeros of a Polynomial

Zero or root of a polynomial f:r a real number for which f(r) =

0r is an x-intercept of the graph

of f.(x { r) is a factor of f.

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Zeros of a Polynomial

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Zeros of a Polynomial

Example.Problem: Find a polynomial of

degree 3 whose zeros are {4, {2 and 3.

Answer:

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Zeros of a Polynomial

Repeated or multiple zero or root of f: Same factor (x { r) appears

more than onceZero of multiplicity m:

(x { r)m is a factor of f and (x { r)m+1 isn’t.

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Zeros of a Polynomial

Example.Problem: For the polynomial, list

all zeros and their multiplicities.f(x) = {2(x { 2)(x + 1)3(x { 3)4

Answer:

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Zeros of a Polynomial

Example. For the polynomialf(x) = {x3(x { 3)2(x + 2)

(a) Problem: Graph the polynomial Answer:

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Zeros of a Polynomial

Example. (cont.)(b) Problem: Find the zeros and

their multiplicities Answer:

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Multiplicity

Role of multiplicity:r a zero of even multiplicity:

f(x) does not change sign at rGraph touches the x-axis at r, but

does not cross

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Multiplicity

Role of multiplicity:r a zero of odd multiplicity:

f(x) changes sign at rGraph crosses x-axis at r

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Turning PointsTurning points:

Points where graph changes from increasing to decreasing function or vice versa

Turning points correspond to local extrema.

Theorem. If f is a polynomial function of degree n, then f has at most n { 1 turning points.

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End Behavior Theorem. [End Behavior]

For large values of x, either positive or negative, that is, for large |x|, the graph of the polynomial

f(x) = anxn + an{1xn{1 + + a1x + a0

resembles the graph of the power function

y = anxn

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End Behavior

End behavior of:f(x) = anxn + an{1xn{1 + + a1x

+ a0

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Analyzing Polynomial Graphs

Example. For the polynomial: f(x) =12x3 { 2x4 { 2x5

(a) Problem: Find the degree. Answer:

(b) Problem: Determine the end behavior. (Find the power function that the graph of f resembles for large values of |x|.)

Answer:

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Analyzing Polynomial Graphs

Example. (cont.)(c) Problem: Find the x-

intercept(s), if any Answer:

(d) Problem: Find the y-intercept. Answer:

(e) Problem: Does the graph cross or touch the x-axis at each x-intercept:

Answer:

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Analyzing Polynomial Graphs

Example. (cont.)(f) Problem: Graph f using a

graphing utilityAnswer:

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Analyzing Polynomial Graphs

Example. (cont.)(g) Problem: Determine the

number of turning points on the graph of f. Approximate the turning points to 2 decimal places.

Answer:

(h) Problem: Find the domain Answer:

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Analyzing Polynomial Graphs

Example. (cont.)(i) Problem: Find the range

Answer:(j) Problem: Find where f is

increasing Answer:

(k) Problem: Find where f is decreasing

Answer:

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Cubic Relations

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Cubic Relations

Example. The following data represent the average number of miles driven (in thousands) annually by vans, pickups, and sports utility vehicles for the years 1993-2001, where x = 1 represents 1993, x = 2 represents 1994, and so on.

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Cubic Relations

Year, x Average Miles Driven, M

1993, 1 12.4

1994, 2 12.2

1995, 3 12.0

1996, 4 11.8

1997, 5 12.1

1998, 6 12.2

1999, 7 12.0

2000, 8 11.7

2001, 9 11.1

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Cubic Relations

Example. (cont.)(a) Problem: Draw a scatter

diagram of the data using x as the independent variable and M as the dependent variable.

Answer:

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Cubic Relations

Example. (cont.)(b) Problem: Find the cubic

function of best fit and graph it

Answer:

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Key Points

Polynomial FunctionsPower FunctionsGraphing Using TransformationsZeros of a PolynomialMultiplicityTurning PointsEnd BehaviorAnalyzing Polynomial GraphsCubic Relations

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The Real Zeros of a Polynomial Function

Section 3.6

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Division Algorithm Theorem. [Division Algorithm]

If f(x) and g(x) denote polynomial functions and if g(x) is a polynomial whose degree is greater than zero, then there are unique polynomial functions q(x) and r(x) such that

where r(x) is either the zero polynomial or a polynomial of degree less than that of g(x).

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Division Algorithm

Division algorithm

f(x) is the dividendq(x) is the quotientg(x) is the divisorr(x) is the remainder

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Remainder Theorem

First-degree divisorHas form g(x) = x { cRemainder r(x)

Either the zero polynomial or a polynomial of degree 0,

Either way a number R.Becomes f(x) = (x { c)q(x) + RSubstitute x = cBecomes f(c) = R

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Remainder Theorem

Theorem. [Remainder Theorem] Let f be a polynomial function. If f(x) is divided by x { c, the remainder is f(c).

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Remainder Theorem

Example. Find the remainder if

f(x) = x3 + 3x2 + 2x { 6 is divided by:(a) Problem: x + 2

Answer:(b) Problem: x { 1

Answer:

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Factor Theorem

Theorem. [Factor Theorem] Let f be a polynomial function. Then x { c is a factor of f(x) if and only if f(c) = 0.

If f(c) = 0, then x { c is a factor of f(x).

If x { c is a factor of f(x), then f(c) = 0.

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Factor Theorem

Example. Determine whether the function

f(x) = {2x3 { x2 + 4x + 3 has the given factor:(a) Problem: x + 1

Answer:(b) Problem: x { 1

Answer:

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Number of Real Zeros

Theorem. [Number of Real Zeros]A polynomial function of degree n, n ¸ 1, has at most n real zeros.

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Rational Zeros Theorem

Theorem. [Rational Zeros Theorem]Let f be a polynomial function of degree 1 or higher of the form

f(x) = anxn + an{1xn{1 + + a1x + a0

an 0, a0 0, where each coefficient is an integer. If p/q, in lowest terms, is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

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Rational Zeros Theorem

Example. Problem: List the potential

rational zeros of

f(x) = 3x3 + 8x2 { 7x { 12

Answer:

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Finding Zeros of a Polynomial

Determine the maximum number of zeros.Degree of the polynomial

If the polynomial has integer coefficients:Use the Rational Zeros Theorem

to find potential rational zeros Using a graphing utility,

graph the function.

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Finding Zeros of a Polynomial

Test values Test a potential rational zeroEach time a zero is found,

repeat on the depressed equation.

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Finding Zeros of a Polynomial

Example. Problem: Find the rational zeros

of the polynomial in the last example.

f(x) = 3x3 + 8x2 { 7x { 12

Answer:

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Finding Zeros of a Polynomial

Example.

Problem: Find the real zeros of

f(x) = 2x4 + 13x3 + 29x2 + 27x +

9

and write f in factored form

Answer:

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Factoring Polynomials

Irreducible quadratic: Cannot be factored over the real numbers

Theorem. Every polynomial function (with real coefficients) can be uniquely factored into a product of linear factors and irreducible quadratic factors

Corollary. A polynomial function (with real coefficients) of odd degree has at least one real zero

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Factoring Polynomials

Example.

Problem: Factor

f(x)=2x5 { 9x4 + 20x3 { 40x2 +

48x {16

Answer:

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Bounds on Zeros

Bound on the zeros of a polynomialPositive number M Every zero lies between {M

and M.

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Bounds on Zeros

Theorem. [Bounds on Zeros]Let f denote a polynomial whose leading coefficient is 1.

f(x) = xn + an{1xn{1 + + a1x + a0

A bound M on the zeros of f is the smaller of the two numbersMax{1, ja0j + ja1j + + jan-1j},

1 + Max{ja0j ,ja1j , … , jan-1j}

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Bounds on Zeros

Example. Find a bound to the zeros of each polynomial.(a) Problem:

f(x) = x5 + 6x3 { 7x2 + 8x { 10 Answer:

(b) Problem: g(x) = 3x5 { 4x4 + 2x3 + x2 +5

Answer:

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Intermediate Value Theorem

Theorem. [Intermediate Value Theorem]Let f denote a continuous function. If a < b and if f(a) and f(b) are of opposite sign, then f has at least one zero between a and b.

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Intermediate Value Theorem

Example. Problem: Show that f(x) = x5 { x4 + 7x3 { 7x2 { 18x +

18 has a zero between 1.4 and 1.5. Approximate it to two decimal places.

Answer:

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Key Points

Division AlgorithmRemainder TheoremFactor TheoremNumber of Real ZerosRational Zeros TheoremFinding Zeros of a PolynomialFactoring PolynomialsBounds on Zeros Intermediate Value Theorem

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Complex Zeros; Fundamental Theorem of Algebra

Section 3.7

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Complex Polynomial Functions

Complex polynomial function: Function of the formf(x) = anxn + an {1xn {1 + + a1x + a0

an, an {1, …, a1, a0 are all complex numbers,

an 0, n is a nonnegative integer x is a complex variable. Leading coefficient of f: an

Complex zero: A complex number r with f(r) = 0.

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Complex Arithmetic

See Appendix A.6. Imaginary unit: Number i with

i2 = {1. Complex number: Number of the

form z = a + bi a and b real numbers. a is the real part of z b is the imaginary part of z

Can add, subtract, multiply Can also divide (we won’t)

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Complex Arithmetic

Conjugate of the complex number a + biNumber a { biWrittenProperties:

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Complex Arithmetic

Example. Suppose z = 5 + 2i and w = 2 { 3i. (a) Problem: Find z + w

Answer:(b) Problem: Find z { w

Answer:(c) Problem: Find zw

Answer:(d) Problem: Find

Answer:

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Fundamental Theorem of Algebra

Theorem. [Fundamental Theorem of Algebra]Every complex polynomial function f(x) of degree n ¸ 1 has at least one complex zero.

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Fundamental Theorem of Algebra

Theorem. Every complex polynomial function f(x) of degree n ¸ 1 can be factored into n linear factors (not necessarily distinct) of the formf(x) = an(x { r1)(x { r2) (x { rn)

where an, r1, r2, …, rn are complex numbers. That is, every complex polynomial function f(x) of degree n ¸ 1 has exactly n (not necessarily distinct) zeros.

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Conjugate Pairs Theorem

Theorem. [Conjugate Pairs

Theorem]

Let f(x) be a polynomial

whose coefficients are real

numbers. If a + bi is a zero of

f, then the complex conjugate

a { bi is also a zero of f.

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Conjugate Pairs Theorem

Example. A polynomial of degree 5 whose coefficients are real numbers has the zeros {2, {3i and 2 + 4i. Problem: Find the remaining

two zeros.Answer:

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Conjugate Pairs Theorem

Example. Problem: Find a polynomial f of

degree 4 whose coefficients are real numbers and that has the zeros {2, 1 and 4 + i.

Answer:

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Conjugate Pairs Theorem

Example. Problem: Find the complex zeros

of the polynomial function f(x) = x4 + 2x3 + x2 { 8x { 20

Answer:

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Key Points

Complex Polynomial Functions

Complex ArithmeticFundamental Theorem of

AlgebraConjugate Pairs Theorem

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Properties of Rational Functions

Section 3.3

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Rational Functions

Rational function: Function of the form

p and q are polynomials,q is not the zero polynomial.

Domain: Set of all real numbers except where q(x) = 0

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Rational Functions

is in lowest terms:The polynomials p and q have

no common factorsx-intercepts of R:

Zeros of the numerator p when R is in lowest terms

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Rational Functions

Example. For the rational function

(a) Problem: Find the domain

Answer:

(b) Problem: Find the x-intercepts

Answer:

(c) Problem: Find the y-intercepts

Answer:

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Graphing Rational Functions

Graph of

-10 -5 5 10

-10

-7.5

-5

-2.5

2.5

5

7.5

10

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Graphing Rational Functions

As x approaches 0, is unbounded in the positive direction. Write f(x) ! 1Read “f(x) approaches infinity”Also:

May write f(x) ! 1 as x ! 0May read: “f(x) approaches

infinity as x approaches 0”

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-6 -4 -2 2 4

-4

-2

2

4

Graphing Rational Functions

Example. ForProblem: Use transformations to

graph f.Answer:

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Asymptotes

Horizontal asymptotes:Let R denote a function.Let x ! {1 or as x ! 1, If the values of R(x) approach

some fixed number L, then the line y = L is a horizontal asymptote of the graph of R.

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Asymptotes

Vertical asymptotes:Let x ! cIf the values jR(x)j ! 1, then the

line x = c is a vertical asymptote of the graph of R.

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Asymptotes

Asymptotes:Oblique asymptote: Neither

horizontal nor vertical Graphs and asymptotes:

Graph of R never intersects a vertical asymptote.

Graph of R can intersect a horizontal or oblique asymptote (but doesn’t have to)

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Asymptotes

A rational function can have:Any number of vertical

asymptotes.1 horizontal and 0 oblique

asymptote0 horizontal and 1 oblique

asymptotes0 horizontal and 0 oblique

asymptotesThere are no other possibilities

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Vertical Asymptotes

Theorem. [Locating Vertical Asymptotes]

A rational function

in lowest terms, will have a

vertical asymptote x = r if r

is a real zero of the

denominator q.

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Vertical Asymptotes

Example. Find the vertical asymptotes, if any, of the graph of each rational function.

(a) Problem:

Answer:

(b) Problem:

Answer:

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Vertical Asymptotes

Example. (cont.)

(c) Problem:

Answer:

(d) Problem:

Answer:

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Horizontal and Oblique Asymptotes

Describe the end behavior of a rational function.

Proper rational function: Degree of the numerator is less

than the degree of the denominator.

Theorem. If a rational function R(x) is proper, then y = 0 is a horizontal asymptote of its graph.

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Horizontal and Oblique Asymptotes

Improper rational function R(x): one that is not proper.May be written

where is proper. (Long division!)

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Horizontal and Oblique Asymptotes

If f(x) = b, (a constant)Line y = b is a horizontal

asymptote If f(x) = ax + b, a 0,

Line y = ax + b is an oblique asymptote

In all other cases, the graph of R approaches the graph of f, and there are no horizontal or oblique asymptotes.This is all higher-degree

polynomials

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Horizontal and Oblique Asymptotes

Example. Find the hoizontal or oblique asymptotes, if any, of the graph of each rational function.

(a) Problem:

Answer:

(b) Problem:

Answer:

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Horizontal and Oblique Asymptotes

Example. (cont.)

(c) Problem:

Answer:

(d) Problem:

Answer:

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Key Points

Rational FunctionsGraphing Rational FunctionsVertical AsymptotesHorizontal and Oblique

Asymptotes

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The Graph of a Rational Function; Inverse and Joint Variation

Section 3.4

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Analyzing Rational Functions

Find the domain of the rational function.

Write R in lowest terms.Locate the intercepts of the

graph.x-intercepts: Zeros of numerator of

function in lowest terms.y-intercept: R(0), if 0 is in the

domain.Test for symmetry – Even, odd or

neither.

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Analyzing Rational Functions

Locate the vertical asymptotes: Zeros of denominator of function

in lowest terms. Locate horizontal or oblique

asymptotes Graph R using a graphing

utility. Use the results obtained to

graph by hand

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Analyzing Rational Functions

Example. Problem: Analyze the graph of

the rational functionAnswer:

Domain: R in lowest terms:x-intercepts:y-intercept: Symmetry:

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Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

Vertical asymptotes: Horizontal asymptote:Oblique asymptote:

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-4 -2 2 4

-4

-2

2

4

Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

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Analyzing Rational Functions

Example. Problem: Analyze the graph of

the rational functionAnswer:

Domain:R in lowest terms:x-intercepts:y-intercept:Symmetry:

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Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

Vertical asymptotes:

Horizontal asymptote:Oblique asymptote:

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-6 -4 -2 2 4 6

-6

-4

-2

2

4

6

Analyzing Rational Functions

Example. (cont.)Answer: (cont.)

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Variation

Inverse variation:Let x and y denote 2

quantities. y varies inversely with x

If there is a nonzero constant such that

Also say: y is inversely proportional to x

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Variation

Joint or Combined Variation:Variable quantity Q

proportional to the product of two or more other variables

Say Q varies jointly with these quantities.

Combinations of direct and/or inverse variation are combined variation.

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Variation

Example. Boyle’s law states that for a fixed amount of gas kept at a fixed temperature, the pressure P and volume V are inversely proportional (while one increases, the other decreases).  

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Variation

Example. According to Newton, the gravitational force between two objects varies jointly with the masses m1 and m2 of each object and inversely with the square of the distance r between the objects, hence

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Key Points

Analyzing Rational FunctionsVariation

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Polynomial and Rational Inequalities

Section 3.5

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Solving Inequalities Algebraically

Rewrite the inequality Left side: Polynomial or rational

expression f. (Write rational expression as a single quotient)

Right side: ZeroShould have one of following

formsf(x) > 0f(x) ¸ 0f(x) < 0f(x) · 0

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Solving Inequalities Algebraically

Determine where left side is 0 or undefined.

Separate the real line into intervals based on answers to previous step.

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Solving Inequalities Algebraically

Test Points:Select a number in each interval Evaluate f at that number.If the value of f is positive, then

f(x) > 0 for all numbers x in the interval.

If the value of f is negative, then f(x) < 0 for all numbers x in

the interval.

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Solving Inequalities Algebraically

Test Points (cont.) If the inequality is strict (< or >)

Don’t include values where x = 0 Don’t include values where x is

undefined.If the inequality is not strict (·

or ¸)Include values where x = 0 Don’t include values where x is

undefined.

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Solving Inequalities Algebraically

Example. Problem: Solve the inequality x5

¸ 16xAnswer:

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Key Points

Solving Inequalities Algebraically