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# Homework, Page 234 - PowerPoint PPT Presentation

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

1.

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

5.

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

9.

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

13.

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

17.

b.

State how many complex and real zeroes the function has.

21.

State how many complex and real zeroes the function has.

25.

From the graph, f has four

complex zeros, two of which

are real.

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

29.

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

33.

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

37.

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

41.

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

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

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.

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.

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

65. Find the three cube roots of 8 by solving

### 2.6

Graphs of Rational Functions

• 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.

• 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

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

Example Finding Asymptotes of Rational Functions

Example Graphing a Rational Function

Use limits to describe the behavior of the function

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

• Review Section: 2.6

• Page 245, Exercises: 1 – 69 (EOO)

### 2.7

Solving Equations in One Variable

• Solving Rational Equations

• Extraneous Solutions

• Applications

… and why

Applications involving rational functions as

models often require that an equation involving

fractions be solved.

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