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Homework, Page 234. Write the polynomial in standard form, and identify the zeros of the function and the x -intercepts of its graph. 1.. Homework, Page 234. Write a polynomial function in minimum degree in standard form with real coefficients whose zeros include those listed. 5..

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Homework, Page 234

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Homework, Page 234

Write the polynomial in standard form, and identify the zeros of the function and the x-intercepts of its graph.

1.


Homework, Page 234

Write a polynomial function in minimum degree in standard form with real coefficients whose zeros include those listed.

5.


Homework, Page 234

Write a polynomial function in minimum degree in standard form with real coefficients whose zeros include those listed.

9.


Homework, Page 234

Write a polynomial function in minimum degree in standard form with real coefficients whose zeros and their multiplicities include those listed.

13.


Homework, Page 234

Match the polynomial function graph to the given zeros and multiplicities.

17.

b.


Homework, Page 234

State how many complex and real zeroes the function has.

21.


Homework, Page 234

State how many complex and real zeroes the function has.

25.

From the graph, f has four

complex zeros, two of which

are real.


Homework, Page 234

Find all zeros and write a linear factorization of the function.

29.


Homework, Page 234

Using the given zero, find all of the zeros and write a linear factorization of f (x).

33.


Homework, Page 234

Write the function as a product of linear and irreducible quadratic factors all with real coefficients.

37.


Homework, Page 234

Write the function as a product of linear and irreducible quadratic factors all with real coefficients.

41.


Homework, Page 234

Answer Yes or No. If yes, give an example; if no, give a reason.

45. Is it possible to find a polynomial of degree 3 with real number coefficients that has -2 as its only real zero.

Yes, for example


Homework, Page 234

Find the unique polynomial with real coefficients that meets these conditions.

49. Degree 4; zeros at x = 3, x = –1, and

x = 2 – i; f (0) = 30


Homework, Page 234

53. There is at least one polynomial with real coefficients with 1 – 2i as its only nonreal zero. Justify your answer.

False, to have real coefficients, the nonreal factors of a polynomial must be in conjugate pairs.


Homework, Page 234

57. Which of the following cannot be the number of nonreal zeros of a polynomial of degree 5 with real coefficients?

a. 0

b. 2

c. 3

d. 4

e. None of the above.


Homework, Page 234

61. Verify that the complex number i is a zero of the polynomial


Homework, Page 234

65. Find the three cube roots of 8 by solving


2.6

Graphs of Rational Functions


Quick Review


Quick Review Solutions


What you’ll learn about

  • Rational Functions

  • Transformations of the Reciprocal Function

  • Limits and Asymptotes

  • Analyzing Graphs of Rational Functions

    … and why

    Rational functions are used in calculus and in scientific

    applications such as inverse proportions.


Rational Functions


Limits

  • We used limits to investigate continuity in Chapter 1

  • Limits may also be used

    • to investigate behavior near vertical asymptotes

    • to investigate behavior as functions approach positive or negative infinity, usually called end behavior


Transformations of the Reciprocal Function

The general form for a function is

In this equation, k indicates units of vertical translation, h indicates units of horizontal translation, and a indicates factor of stretch.

For instance,

indicates the reciprocal

function is translated 4

units left and 3 down and

stretched by a factor of 3.


Example Finding the Domain of a Rational Function


Graph a Rational Function


Graph a Rational Function (Cont’d)


Example Finding Asymptotes of Rational Functions


Example Graphing a Rational Function


Analyzing Graphs

Use limits to describe the behavior of the function


Finding Properties of Graphs

Find the intercepts, vertical asymptotes, end behavior asymptotes, and graph the function.


Homework

  • Review Section: 2.6

  • Page 245, Exercises: 1 – 69 (EOO)


2.7

Solving Equations in One Variable


Quick Review


Quick Review Solutions


What you’ll learn about

  • Solving Rational Equations

  • Extraneous Solutions

  • Applications

    … and why

    Applications involving rational functions as

    models often require that an equation involving

    fractions be solved.


Extraneous Solutions

When we multiply or divide an equation by an expression containing variables, the resulting equation may have solutions that are not solutions of the original equation. These are extraneous solutions. For this reason we must check each solution of the resulting equation in the original equation.


Example Solving by Clearing Fractions


Example Solving a Rational Function


Example Eliminating Extraneous Solutions


Example Eliminating Extraneous Solutions


Example Finding an Acid Solution


Example Finding an Acid Solution


Example Finding an Acid solution


Example Finding an Acid Solution


Example Finding a Minimum Perimeter


Example Finding a Minimum Perimeter


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