mat 129 – chapter 5 slides 129 - chapter 5...mat 129 ‐chapter 5 8 rational functions example:...

MAT 129 ‐ Chapter 5

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MAT 129 – PrecalculusChapter 5 NotesPolynomial and Rational FunctionsDavid J. Gisch

Polynomial Functions and Models

Polynomial Functions

The degree of a polynomial is the highest power of that polynomial, which is using the above notation. If is even we also say the polynomial has an even degree.If is odd we say the polynomial has an odd degree.

Polynomial FunctionsExample: Determine which of the following are polynomial functions. For those that are, state the degree.

a) 2 4

b) 4 2 7

c) 3 400

d) 20

e) 2 2 3

f)


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Polynomial FunctionsThe graph of polynomial functions must be smooth and continuous.

Polynomial Functions

Polynomial FunctionsExample: What happens when the degree of a power function is even?

Polynomial FunctionsExample: What happens when the degree of a power function is odd?


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Polynomial FunctionsWhat happens when the degree of a power function is even or odd?

Polynomial FunctionsWe can still use transformations to graph polynomial functions, to a certain degree (LOL).

Polynomial FunctionsWe can still use transformations to graph polynomial functions, to a certain degree (LOL).

Polynomial FunctionsIf a polynomial is not a power function we can still say much about it by looking at the degree. The degree and the leading coefficient tell us the “end” behavior.

Larger exponents (powers) dominate smaller exponents when inputs get large.

The leading term (the term with the highest power) is going to dominate and be more important than all remaining terms.

1 2 3 4 5 6

1 4 9 16 25 361 8 27 64 125 2161 16 81 256 625 1296

4 3 4 48


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The Degree of a PolynomialThe degree tells us how many possible times the graph touches/crosses the x-axis. • It touches or crosses the -axis at most times.• If it is an even degree it may not cross/touch the x-axis at all. If it is an

odd degree, it must cross the x-axis at least once.The degree tells us the maximum possible turning points. The number maximum number of turning points is one less than the degree.• It has at most -1 turns.

a) 3 400

b) 2 2 3

Polynomial FunctionsExample: For each graph state the minimum possible degree of the polynomial and whether that degree is even or odd.

Zeros/Roots of Polynomials Zeros/Roots of PolynomialsIf is a factor of a polynomial, then we say is a root of multiplicity .


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Zeros, Turns, End, and Degree

1 3 4 24 9 40 4 48

Polynomial FunctionsExample: Give me an equation of a polynomial with roots of 3, 4, and2.

Is that the only answer?????

Polynomial FunctionsExample: Give me an equation of a polynomial with roots of 0, 1, and 2.

Polynomial FunctionsExample: Give me an equation of a polynomial that touches the x-axis at 4, crosses it at 3 and 5, has 5 turning points, and has an even degree.


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Graphing Polynomials Polynomial FunctionsExample: Graph the polynomial function

4 3

Polynomial FunctionsExample: Graph the polynomial function

1 5 x 3

Polynomial FunctionsExample: Graph the polynomial function

3 2 3


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Properties of Rational Functions

Factoring (Over the Integers)If it is of the form• where b is a perfect square then it can be factored as two

conjugates.9 3 3

• it cannot be factored.

• , might or might not factor. Recall that the two factors must add to and multiply to .

12 4 3

• , might or might not factor. There are several methods to do this. I try quick trial and error.

Factoring QuadraticsExample: Factor (over the integers).

a) 2 8

b) 8 2

c) 2 32

d) 5 24

Rational FunctionsA rational function is a function of the form

Where and are polynomial functions, and 0.

Note: Since the function is in the form of a fraction we need to be mindful of values that make the denominator undefined.


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Rational FunctionsExample: Find the domain of each rational function.

a)

b) 1

c)

Rational FunctionsWe often have horizontal and vertical asymptotes when graphing rational functions. Recall that an asymptote is an imaginary boundary that the graph gets close to but never quite crosses.

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Types of AsymptotesHorizontal Asymptotes

VerticalAsymptotes

ObliqueAsymptotes

Rational FunctionsA rational function , in lowest terms, will have a vertical asymptote at if x is a factor of the denominator .When we say “lowest terms” we mean that the rational function has been simplified.

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Rational FunctionsIf a rational function does have a factor in the denominator that cancels there is a still a domain issue and it creates what we call a hole at if x is the canceled factor of the denominator.

2 33 2

There is an asymptote at 2, and there is a hole at 3. The hole will not

show up on your calculator’s graph.

Rational Functions• Graph the function

2 32


3 24 20

• Try to factor the denominator. I don’t see an easy answer, so use the quadratic equation.

4 16 4 1 20 64 0• So this means there are no REAL factors of the

denominator. So no vertical asymptotes.

Rational FunctionsExample: Find the vertical asymptotes.

a)

b)

c)


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Rational FunctionsExample: Create a rational function that has a vertical asymptote at 3 and 4, and has a hole at 6.

Rational FunctionsA rational function is called proper if the degree of the numerator is less than the degree of the denominator.

If a rational function is proper, the line 0 is a horizontal asymptote.

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12

Rational Functions

11 2


3 24 20

• Try to factor the denominator. I don’t see an easy answer, so use the quadratic equation.

4 16 4 1 20 64 0• So this means there are no REAL factors of the

denominator. So no vertical asymptotes.

But it is proper!


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Rational FunctionsA rational function is called improper if the degree of the numerator is greater than or equal to the degree of the denominator.

If a rational function is improper, we must factor the polynomial into the sum of a polynomial and rational function.

8 24 1 2

44 1

Horizontal asymptote at y=2.

Rational FunctionsExample: Find the “horizontal” asymptote.

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Rational FunctionsExample: Find the “horizontal” asymptote.

2 21

Rational FunctionsExample: Find all the asymptotes.

3 43


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Rational FunctionsExample: Find all the asymptotes.

85 6

The Graph of a Rational Function

Graphing Rational FunctionsExample: Give a possible equation for the graph.



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Graphing Rational FunctionsExample: Graph the rational function.

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1


14


1


14


6 123 5 2

Polynomial and Rational Inequalities


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Steps for Solving Inequalities Solving Rational InequalitiesExample: Solve the inequality 4 , and graph the solution set.

1. Bring all of the terms to the left so that zero is on the right. Divide by 1 if the leading coefficient is negative.

2. Find the zeros (roots).

3. Use the numbers found in 2 to separate the number line into intervals.

Example 5.4.1 Continued4. Select a number in each interval and evaluate f at that

number.

Solving Rational InequalitiesExample: Solve the inequality and graph the solution set.

3 21 0

1. Bring all of the terms to the left so that zero is on the right. 2. Find the zeros (roots) of both the numerator and the denominator.

3. Use the numbers found in 2 to separate the number line into intervals.


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Example Continued4. Select a number in each interval and evaluate f at that

number.

3, 1 ∪ 1, 2

Solving Rational InequalitiesExample Solve the inequality and graph the solution set.

4 52 3

Solving Rational InequalitiesExample Solve the inequality and graph the solution set.

2 15 0

The Real Zeros of a Polynomial Function


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The Division Algorithm The Division AlgorithmThe division algorithm of polynomials is the same as the one for integers.

The Remainder & Factor TheoremsRemainder TheoremLet be a polynomial function. If is divided by , then the remainder is .

Factor TheoremLet be a polynomial function. Then is a factor of

if and only if 0.

The Remainder TheoremFor example, let’s say I have a hunch that 1 is a factor of 8. In other words

8 1 ? ? ? ?

Using the remainder theorem, we calculate1 1 8 1 8 9

Thus, as there is a remainder ( 0) we know 1 does not divide evenly (i.e. it is not a factor).


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Factor TheoremSuppose I now try 2 as a factor of 8. In other words

8 2 ? ? ? ?

Using the remainder theorem, we calculate2 2 8 8 8 0

The remainder is zero, so my hunch is correct and 2is indeed a factor. It remains for us to find (????) from above but now we know it is a factor.

If there is no remainder, how do we find the remaining factor?• Use long division, or• Synthetic division

Long Division

2 0 0 8

Synthetic DivisionTo understand synthetic division let us use the previous example of testing 2 as a factor of 8. In other words

8 2 ? ? ? ?

100 82

1240

248

8 2 2 4

No Remainder, so it is a factor.

Remainder/Factor TheoremsExample Use synthetic division to complete the following operation.

2 3 42


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Remainder/Factor TheoremsExample Determine if the following are factors of

2 4 3If it is, follow up with synthetic division and factor it.

a) 1

b) 1

Remainder/Factor TheoremsExample Use synthetic or long division to find all of the asymptotes of

2 5 63 4

Find the Real ZerosLet be a polynomial function of degree , with 1. Then has at most real zeros.

So there are at most n but how can we tell how many for sure?• Descartes’ rule of signs.• Rational root theorem.


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Decartes’ Rule of Signs Remainder/Factor TheoremsExample Find the number of possible real zeros of

3 4 3 2 3

Positives: 3 4 3 2 3

Negatives: 3 4 3 2 3

Remainder/Factor TheoremsExample Find the number of possible real zeros of

8 18 11

Rational Root TheoremSo we know many real zeros from Decartes’ theorem but we still don’t know what they are. We use the rational root theorem for that.


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Remainder/Factor TheoremsExample Use the rational root theorem to find the rational roots of

3 4 3 2 3

“Work smarter, not harder!”

Candidates: , , , 1, , 3: 1, 3: 1, 3

1 3 0 -4 3 2 -1 -3

Remainder/Factor TheoremsExample Use the rational root theorem and Descartes's theorem to find the rational roots of

8 18 11

Summary


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Remainder/Factor TheoremsExample Find the real zeros of

2 13 29 27 9.Use the factors you found to rewrite the polynomial in factored form.

Factoring Polynomials

Every polynomial with real coefficients can be uniquely factored into a product of linear and/or irreducible quadratic functions.

4 2 3 10

A Polynomial, with real coefficients ,of odd degree has at least one real zero.

This is because the end behavior is opposite so it must cross the graph (i.e. have a REAL zero).

Bounds on Zeros


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Bounds on ZerosExample Find the bounds on the zeros of

2 13 29 27 9.

Bounds on ZerosExample Find the bounds on the zeros of

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58

37

29 .

Intermediate Value TheoremLet be a polynomial function. If and and

are of the opposite sign, then there is at least one real zero of between and .

IVTExample Show that 1 has a zero between1and 2.


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Summary• Find all of the indicated answers on your handout.

Complex Zeros; Fundamental Theorem of Algebra

Complex Numbers• Recall that a complex number is always of the form

where and are real numbers.

• Further, 1.

• Factor 3 10

Complex Polynomials

4 3 2

7 4 12 8


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Complex Polynomials

Every complex polynomial function of degree 1has at least one complex zero.

Complex Polynomials

Let be a polynomial whose coefficients are real numbers. If is a zero of , the complex conjugate ̅ is a zero of (i.e. they always come in pairs).

Complex PolynomialsExample You are told a polynomial of degree 5 whose coefficients are real numbers has the zeros 2, 3 , and 2 4 .

a) Find the remaining two zeros.

b) Use this info to write the function as a product of linear factors and/or irreducible quadratic functions.


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Complex PolynomialsExample You are told a polynomial of degree 4 whose coefficients are real numbers has the zeros 1, 1, and 4 .

a) Find the remaining zero(s).

b) Use this info to write the function as a product of linear factors and/or irreducible quadratic functions.

SummaryExample Use all these resources to factor

2 8 20

mat 129 – chapter 5 slides 129 - chapter 5...mat 129 ‐chapter 5 8 rational functions example:...

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