finding real roots of polynomial equations · finding real roots of polynomial equations extension:...

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Name Class Date 4-1

Understanding Polynomial Functions

A polynomial function can be written as f (x) = a n x n + a n - 1 x n - 1 + … + a 1 x + a 0 where the coefficients a n , . . ., a 1 , a 0 are real numbers. This is known as standard form. Note that linear functions are polynomial functions of degree 1 and quadratic functions are polynomial functions of degree 2. The functions you have graphed in the form f (x) = a(x - h) n + k are also polynomial functions, as you will see in later lessons.

The end behavior of a polynomial function f(x) = a n x n + a n - 1 x n - 1 + … + a 1 x + a 0 is determined by the term with the greatest degree, a n x n .

You know how to use transformations to help you graph functions of the form f (x) = a (x - h) n + k. If a polynomial function is not written in this form, then other graphing methods must be used. You will learn several methods throughout this unit, but the most basic is to plot points and connect them with a smooth curve, taking into account the end behavior of the function.

E n g a g E1

Finding Real Roots of Polynomial EquationsExtension: Graphing Factorable Polynomial FunctionsEssential question: How do you use zeros to graph polynomial functions?

Polynomial End Behavior

n a n As x → +∞ As x → -∞ Graph

Even Positive f(x) → +∞ f(x) → +∞

Even Negative f(x) → -∞ f(x) → -∞

Odd Positive f(x) → +∞ f(x) → -∞

Odd Negative f(x) → -∞ f(x) → +∞

prep for MCC9–12.F.IF.7c

Video Tutor

Module 4 109 Lesson 1

Degree 1Linear

Degree 2Quadratic

Degree 3Cubic

Degree 4Quartic

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In general, the higher the degree of a polynomial, the more complex the graph. Here are some typical graphs for polynomials of degree n.

REFLECT

1a. Evaluate each term of the polynomial f (x) = x 4 + 2 x 3 - 5 x 2 + 2x - 3 for x = 10 and x = -10. Explain why the end behavior is determined by the term with the greatest degree.

1b. Describe the end behavior of f (x) = -2 x 3 + 3 x 2 - x - 3. Explain your reasoning.

The nested form of a polynomial has the form f (x) = (((ax + b)x + c)x + d)x ...). It is often simpler to evaluate polynomials in nested form.

Writing Polynomials in Nested Form

Write f(x) = x 3 + 2 x 2 - 5x - 6 in nested form.

f (x) = x 3 + 2 x 2 - 5x - 6 Write the function.

f (x) = ( ) x - 6 Factor an x out of the first three terms.

f (x) = (( )x - 5)x - 6 Factor an x out of the first two terms in parentheses.

REFLECT

2a. Use the standard form and the nested form to evaluate the polynomial for x = 3. Then compare both methods. Which is easier? Why?

E X a M P L E2MCC9–12.F.IF.2


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Graphing Polynomials

Graph f(x) = x 3 + 2 x 2 - 5x - 6.

A Determine the end behavior of the graph. The end behavior is determined by the leading term, x 3 .

So, as x → +∞, f (x) → , and as x → -∞, f (x) → .

B Complete the table of values. Use the nested form from the previous example to evaluate the polynomial.

C Plot the points from the table on the graph, omitting any points whose y-values are much greater than or much less than the other y-values on the graph.

D Draw a smooth curve through the plotted points, keeping in mind the end behavior of the graph.

REFLECT

3a. What are the zeros of the function? How can you identify them from the graph?

3b. What are the approximate values of x for which the function is increasing? decreasing?

3c. A student wrote that f (x) has a minimum value of approximately -8. Do you agree or disagree? Why?

3d. Without graphing, what do you think the graph of g (x) = - x 3 - 2 x 2 + 5x + 6 looks like? Why?

E X a M P L E3

x -3 -2 -1 0 1 2 3

f(x)

MCC9–12.F.IF.7c


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Investigating the Behavior of Graphs Near Zeros

Use a graphing calculator to graph each function. Sketch the graphs on the axes provided below. Then complete the table.

E X P L O R E4

A f (x) = (x - 1)(x - 2)(x - 3)(x - 4)

C h(x) = (x - 1 ) 3 (x - 2)

E

B g(x) = (x - 1) 2 (x - 2)(x - 3)

D j(x) = (x - 1 ) 4

Now you will sketch a variety of polynomial functions. You do not need to put values on the y-axis. The emphasis is on showing the overall shape of the graph and its x-intercepts.

Examining Zeros f(x) g(x) h(x) j(x)

What are the zeros of the function? 1, 2, 3, 4

How many times does each zero occur in the factorization?

1: 2 times2, 3: 1 time

At which zero(s) does the graph cross the x-axis? 1, 2

At which zero(s) is the graph tangent to the x-axis? 1

MCC9–12.F.IF.7c

REFLECT

4a. Based on your results, make a generalization about the number of times a zero occurs in the factorization of a function and whether the graph of the function crosses or is tangent to the x-axis at that zero.


-4 -2 2 40

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The factored form of a polynomial is useful for graphing because the zeros can easily be determined. The degree of a polynomial in factored form is the sum of the degrees of the factors.

Sketching the Graph of a Factored Polynomial Function

Sketch the graph of f(x) = (x + 2 ) 2 (x + 1)(x - 2)(x - 3).

A Determine the end behavior.

The degree of the polynomial is the sum of the degrees of the factors.

So, the degree of f (x) is .

If you multiply the factors to write f (x) standard form, a n x n + a n - 1 x n - 1 + … + a 1 x + a 0 , the leading coefficient a n is .

Because the degree is odd and the leading coefficient is positive,

f (x) → as x → +∞ and f (x) → as x → -∞.

B Describe the behavior at the zeros.

The zeros of the function are .

Identify how many times each zero occurs in the factorization.

Determine the zero(s) at which the graph crosses the x-axis.

Determine the zero(s) at which the graph is tangent to the x-axis.

C Sketch the graph at right. Use the end behavior to determine where to start and end. You may find it helpful to plot a few points between the zeros to help get the general shape of the graph.

REFLECT

5a. Can you determine how many times a zero occurs in the factorization of a polynomial function just by looking at the graph of the function? Explain.

E X a M P L E5MCC9–12.F.IF.7c


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The Rational Zero Theorem

If you multiply the factors of the function f (x) = (x - 1)(x - 2)(x - 3)(x - 4), you can write f (x) in standard form as follows.

f (x) = x 4 - 10 x 3 + 35 x 2 - 50x + 24

So, f (x) is a polynomial with integer coefficients that begins with the term x 4 and ends with the term 24. The zeros of f (x) are 1, 2, 3, and 4. Notice that each zero is a factor of the constant term, 24.

Now consider the function g(x) = (2x - 1)(3x - 2)(4x - 3)(5x - 4). If you multiply the factors, you can write g(x) in standard form as follows.

g(x) = 120 x 4 - 326 x 3 + 329 x 2 - 146x + 24

So, g(x) is a polynomial with integer coefficients that begins with the term 120 x 4 and ends

with the term 24. The zeros of g(x) are 1 __ 2 , 2 __ 3 , 3 __ 4 , and 4 __ 5 . In this case, the numerator of each

zero is a factor of the constant term, 24, and the denominator of each zero is a factor of

the leading coefficient, 120.

These examples illustrate the Rational Zero Theorem.

REFLECT

6a. If c __ b

is a rational zero of a polynomial function p(x), explain why bx - c must be

a factor of the polynomial.

Using the Rational Zero Theorem

Sketch the graph of f(x) = x 3 - x 2 - 8x + 12.

A Use the Rational Zero Theorem to identify the possible rational zeros of f (x).

The constant term is 12.

Integer factors of the constant term are ±1, ±2, ±3, ±4, ±6, and ±12.

The leading coefficient is .

Integer factors of the leading coefficient are .

E n g a g E6

E X a M P L E7

Rational Zero Theorem

If p(x) = a n x n + a n - 1 x n - 1 + … + a 2 x 2 + a 1 x + a 0 has integer coefficients, then every rational zero of p(x) is a number of the following form:

c __ b

= factor of constant term a 0

________________________ factor of leading coefficient a n

prep for MCC9–12.A.APR.3

MCC9–12.A.APR.3


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By the Rational Zero Theorem, the possible rational zeros of f (x) are all rational

numbers of the form c __ b

where c is a factor of the constant term and b is a factor

of the leading coefficient.

List all the possible rational zeros.

Possible rational zeros:

B Test the possible rational zeros until you find one that is an actual zero.

Use synthetic substitution to test 1 and 2.

So, is a zero, and therefore is a factor of f (x).

C Factor f (x) = x 3 - x 2 - 8x + 12 completely.

Use the results of the synthetic substitution to write f (x) as the product of a linear factor and a quadratic factor.

f (x) = ( )( )

Factor the quadratic factor to write f (x) as a product of linear factors.

f (x) = ( )( )( )

Use the factorization to identify the other zeros of f (x).

How many times does each zero occur in the factorization?

D Determine the end behavior.

f (x) → as x → +∞ and f (x) → as x → -∞.

E Sketch the graph of the function on the coordinate plane at right.

REFLECT

7a. How did you determine where the graph crosses the x-axis and where it is tangent to the x-axis?

1 -1 -8 12

1 -1 -8 12


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7b. How did factoring the polynomial help you graph the function?

7c. How did using the Rational Zero Theorem to find one zero help you find the other zeros?

P R a c t i c E

Write each polynomial function in nested form. Then sketch the graph by plotting points and using end behavior.

1. f(x) = x 4 - 4 x 2

f(x) =

3. f(x) = x 3 - x 2 - 4x + 4

f(x) =

2. f(x) = - x 3 + 4 x 2 - x - 6

f(x) =

4. f(x) = - x 4 + 4 x 3 - 2 x 2 - 4x + 3

f(x) =


82 4 6

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5. Given the graph of a polynomial function, how can you tell if a given zero occurs an even or an odd number of times?

10. From 2000 to 2010, the profit (in thousands of dollars) for a small business is modeled by P(x) = - x 3 + 9 x 2 - 6x - 16, where x is the number of years since 2000.

a. Sketch a graph of the function at right.

b. What are the zeros of the function in the domain 0 ≤ x ≤ 10?

c. What do the zeros represent?

Sketch the graph of each factored polynomial function.

6. f (x) = (x - 3)(x + 2 ) 2

8. h(x) = (x + 3)(x + 1) 2 (x - 1)

7. g(x) = (x + 1 ) 6

9. j(x) = (x + 2) 3 (x - 3 ) 2


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Use the Rational Zero Theorem to identify the possible zeros of each function. Then factor the polynomial completely. Finally, identify the actual zeros and sketch the graph of the function.

10. f (x) = x 3 - 2 x 2 - x + 2

Possible zeros:

Factored form of function:

Actual zeros:

11. g(x) = x 3 - 2 x 2 - 11x + 12

Possible zeros:


Actual zeros:

12. h(x) = 2 x 4 - 5 x 3 - 11 x 2 + 20x + 12

Possible zeros:


Actual zeros:

13. The polynomial function p(x) has degree 3, and its zeros are -3, 4, and 6. What do you think is the equation of p(x)? Do you think there could be more than one possibility? Explain.


Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Algebra 2

Practice Finding Real Roots of Polynomial Equations

Solve each polynomial equation by factoring. 1. 9x

3 − 3x 2 − 3x + 1 = 0 2. x

5 − 2x 4 − 24x

3 = 0

_________________________________________ ________________________________________

3. 3x 5 + 18x

4 − 21x 3 = 0 4. −x

4 + 2x 3 + 8x

2 = 0

_________________________________________ ________________________________________

Identify the roots of each equation. State the multiplicity of each root. 5. x

3 + 3x 2 + 3x + 1 = 0 6. x

3 + 5x 2 − 8x − 48 = 0

_________________________________________ ________________________________________

Identify all the real roots of each equation. 7. x

3 + 10x 2 + 17x = 28 8. 3x

3 + 10x 2 − 27x = 10

_________________________________________ ________________________________________

Solve. 9. An engineer is designing a storage compartment in a spacecraft. The

compartment must be 2 meters longer than it is wide and its depth must be 1 meter less than its width. The volume of the compartment must be 8 cubic meters.

a. Write an equation to model the volume of the compartment.

_________________________________________________________________________________________

b. List all possible rational roots.

c. Use synthetic division to find the roots of the polynomial equation. Are the roots all rational numbers?

_________________________________________________________________________________________

d. What are the dimensions of the storage compartment?

18

LESSON

3-5

CS10_A2_MEPS709987_C03PWBL05.indd 18 4/21/11 4:52:57 PM

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4-1Name Class Date

Additional Practice


Name ________________________________________ Date __________________ Class__________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Algebra 2

Problem Solving Finding Real Roots of Polynomial Equations

Most airlines have rules concerning the size of checked baggage. The rules for Budget Airline are such that the dimensions of the largest bag cannot exceed 45 in. by 55 in. by 62 in. A designer is drawing plans for a piece of luggage that athletes can use to carry their equipment. It will have a volume of 76,725 cubic inches. The length is 10 in. greater than the width and the depth is 14 in. less than the width. What are the dimensions of this piece of luggage? 1. Write an equation in factored form to model the volume of the piece of luggage.

_________________________________________________________________________________________

2. Multiply and set the equation equal to zero.

_________________________________________________________________________________________

3. Think about possible roots of the equation. Could a root be a multiple

of 4? ______________ a multiple of 5? ______________

a multiple of 10? ______________. How do you know?

_________________________________________________________________________________________

4. Use synthetic substitution to test possible roots. Choose positive integers that are factors of the constant term and reasonable in the context of the problem.

Possible Root 1 −4 −140 −76,725

_________________________________________________________________________________________

Choose the letter for the best answer.

5. Which equation represents the factored polynomial? A (w + 55)(w

2 + 25w + 1550) = 0 B (w − 35)(w

2 + 60w + 1405) = 0 C (w − 45)(w

2 + 41w + 1705) = 0 D (w − 4)(w

2 − 140w + 76,725) = 0

6. Which could be the dimensions of this piece of luggage? A 31 in. by 45 in. by 55 in. B 45 in. by 55 in. by 55 in. C 45 in. by 45 in. by 55 in. D 45 in. by 55 in. by 62 in.

100

LESSON

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CS10_A2_MEPS709987_C03PSL05.indd 100 4/21/11 4:52:52 PM

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Problem Solving


finding real roots of polynomial equations · finding real roots of polynomial equations extension:...

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