evaluate and graph polynomial functions -...

Copyright © McDougal Littell/Houghton Mifflin Company. Georgia Notetaking Guide, Mathematics 3 75

2.1 Evaluate and Graph Polynomial FunctionsGoal p Evaluate and graph polynomial functions.Georgia

PerformanceStandard(s)

MM3A1b, MM3A1c, MM3A1d

Your Notes

VOCABULARY

Polynomial

Polynomial function

Degree of a polynomial function

Leading coefficient

Standard form of a polynomial function

Synthetic substitution

End behavior

Even function

Odd function

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76 Georgia Notetaking Guide, Mathematics 3 Copyright © McDougal Littell/Houghton Mifflin Company.

Your Notes

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree and leading coefficient.

a. f (x) 5 3x3 1 4x2.5 2 6x2 b. f (x) 5 x2 1 3.7x 1 9x4

Solution

a. The function a polynomial function because the term has an exponent that is

.

b. The function a polynomial function written as in its standard form.

It has degree and a leading coefficient of .

Example 1 Identify polynomial functions

1. State the degree and leading coefficient of f (x) 5 22x3 1 2x2 2 3x4 1 5.

Checkpoint Complete the following exercise.

Use synthetic substitution to evaluate f(x) 5 2x4 1 3x3 2 6x2 1 3 when x 5 2.

Write the coefficients of f (x) in order of exponents. Write the value of x to the left. Bring down the leading coefficient. Multiply the leading coefficient by

and write the product under the second coefficient. . Multiply the previous sum by and write the

product under the third coefficient. Add. Repeat for all of the remaining coefficients.

2 3 26 0 3 coefficients

f (2) 5

Example 2 Evaluate by synthetic substitution

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Your NotesEND BEHAVIOR OF POLYNOMIAL FUNCTIONS

For the graph of f (x) 5 anxn 1 an 2 1xn 2 1 1 . . . 1 a1x 1 a0:

• If n is odd and an > 0, then f (x) → as x → 1ànd f (x) → as x → 2`.

• If n is odd and an < 0, then f (x) → as x → 1ànd f (x) → as x → 2`.

• If n is even and an > 0, then f (x) → as x → 1ànd f (x) → as x → 2`.

• If n is even and an < 0, then f(x) → asx → 1` and f (x) → as x → 2`.

Example 3 Graph and analyze a polynomial function

(a) Graph the function f(x) 5 2x3 1 2x2 1 2x 2 1, (b) find the domain and the range of the function, (c) describe the degree and leading coefficient of the function, and (d) decide whether the function is even, odd, or neither and describe any symmetries in the graph.

Solutiona. Make a table of values and plot

the corresponding points. Connect the points with a smooth curve.

x 21 0 1 2 3

f(x)

y

x

y

1

1

b. The domain is and the range is .

c. The degree is and the leading coefficient is .

d. The function is because

f(2x) 5 2(2x)3 1 2(2x)2 1 2(2x) 2 1

5

which is not equal or . The graph has .

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Your Notes

Homework

2. Evaluate f (x) for x 5 22 using synthetic substitution.

3. Graph f(x).

y

x

y

1

1

Checkpoint Complete the following exercises using the function f(x) 5 2x4 1 3x3 1 x2 2 4x 2 1.

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Name ——————————————————————— Date ————————————

LESSON

2.1 PracticeDecide whether the function is a polynomial function. If it is, write the function in standard form.

1. f (x) 5 5x 1 2 2. f (x) 5 2x

3. g(x) 5 15 1 3x2 1 x 4. h(x) 5 1 }

2 x4 2 x2 1 3x3

State the degree and leading coeffi cient of the polynomial.

5. g(x) 5 2x2 2 4x 1 9 6. h(x) 5 7 2 3x

7. f (x) 5 2 3 } 4 x3 1 2x4 1 7 8. g(x) 5 8x 2 6 1 x3 Ï

}

2

Use direct substitution to evaluate the polynomial function for the given value of x.

9. f (x) 5 3x3 1 4x2 2 5x 1 7; x 5 1 10. g(x) 5 x5 2 2x 1 3x2 2 9; x 5 2

11. h(x) 5 4x 2 8x6 1 1; x 5 0 12. f (x) 5 x 1 2x3 2 x4 2 10; x 5 22

Use synthetic substitution to evaluate the polynomial function for the given value of x.

13. g(x) 5 2x3 2 5x2 1 4x 2 1; x 5 1 14. h(x) 5 x4 1 7x3 1 x2 2 2x 2 6; x 5 23



Use what you know about end behavior to match the polynomial function with its graph.

15. f (x) 5 2x4 1 2x 2 1 16. f (x) 5 2x2 1 3x 2 2 17. f (x) 5 2x3 1 x2 2 1

A.

x

y

1

1

B.

x

y

1

1

C.

x

y

1

1

Decide whether the function is even, odd, or neither. Describe any symmetries in the graph.

18. f (x) 5 x3 19. f (x) 5 2x4 20. g(x) 5 x3 1 1

21. Business The cost of manufacturing a product can be

0 10 205 15 25 n0

250

500

750

10001250

1500

1750C

Units (thousands)

Co

st

(do

llars

)

modeled by the function C(n) 5 0.2n3 2 7n2 1 108n 1 100 where C is the cost in dollars and n is the number of units of the product in thousands.

a. State the degree of the function.

b. Complete the table of values for the function.

n 0 5 10 15 20

C

c. Use your table to graph the function.

LESSON

2.1 Practice continued

Name ——————————————————————— Date ————————————



2.2 Translate Graphs of Polynomial Functions

GeorgiaPerformanceStandard(s)

MM3A1a, MM3A1c, MM3A1d

Your Notes

Goal p Graph translations of polynomial functions.

1. Graph g(x) 5 x4 2 1. Compare the graph with the graph of f(x) 5 x4.

x

y

1

1

f


Example 1 Translate a polynomial function vertically

Graph g(x) 5 x4 1 1. Compare the graph with the graph of f(x) 5 x4.

1. Make a table of values and plot thecorresponding points.

x 22 21 0 1 2

y x

y

1

1

f

2. Connect the points with a smooth curve and check the end behavior. The degree is and the leading coefficient is . So, g(x) → as x → and g(x) → as x → .

3. Compare with f(x) 5 x4. The graph of g(x) 5 x4 1 1 is the graph of f(x) 5 x4 translated up unit. The domains of f and g are . The range of f is y ≥ 0 and the range of g is .The function f has x- and y-intercepts of 0 and g has a y-intercept of . Notice that both f and g are symmetric with respect to the and are functions because f(2x) 5 (2x)4 5 and g(2x) 5 (2x)4 1 1 5 .

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Your Notes Example 2 Translate a polynomial function horizontally

Graph g(x) 51}2(x 1 2)3. Compare the graph with the

graph of f(x) 51}2x3.


x 24 22 0 2 4

y

x

y

2

2

f


3. Compare with f(x) 51}2x3. The graph of

g(x) 51}2 (x 1 2)3 is the graph of f(x) 5

1}2 x3 translated

to the left units. The domains and ranges of f and g are . The function f has x- and y-intercepts of 0 and g has an x-intercept of and a y-intercept of . Notice that f is symmetric with respect to the and is an function

because f(2x) 51}2(2x)3 5 . Notice that g

is because g(2x) 5

1}2(2x 1 2)3, which is not equal to or .

2. Graph g(x) 5 2(x 2 3)3. Compare the graph with the graph of f(x) 5 2x3.

2

2

f

x

y

2

2

f


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Your Notes

3. Graph g(x) 5 2(x 1 4)3 2 1. Compare the graph with the graph of f(x) 5 2x3.

x

y

1

21

f


Homework

Example 3 Translate a polynomial function

Graph g(x) 5 2(x 2 2)4 1 3. Compare the graph with the graph of f(x) 5 2x4.


x 0 1 2 3 4

y

x

y

1

2

f


3. Compare with f(x) 5 2x4. The graph of g(x) 5 2 (x 2 2)4 1 3 is the graph of f(x) 5 2x4

translated units and .The domains of f and g are . The range of f is and the range of g is .The function f has x- and y-intercepts of 0 and g has x-intercepts of about and and a y-intercept of . Notice that f is symmetric with respect to the and is an function because f(2x) 5 2(2x)4 5 . Notice that gis because g(2x) 5 2(2x 2 2)4 1 3, which is not equal to

or .

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Match the function with its graph.

1. y 5 x3 1 1 2. y 5 x3 3. y 5 (x 1 1)3

A.

x

y

1

1

B.

x

y

1

1

C.

x

y

2

1

Explain how the graphs of f and g are related.

4. f (x) 5 x3, g(x) 5 x3 1 2 5. f (x) 5 x4, g(x) 5 (x 2 2)4

6. f (x) 5 2x3, g(x) 5 2x3 2 4 7. f (x) 5 2 x4, g(x) 5 2 (x 1 3)4

Graph the function. Compare the graph with the graph of f (x) 5 x3.

8. g(x) 5 (x 2 1)3 9. g(x) 5 x3 1 2

x

y

1

1

f

x

y

1

1

f

LESSON

2.2 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Graph the function. Compare the graph with the graph of f (x) 5 x4.

10. g(x) 5 x4 2 4 11. g(x) 5 (x 1 2)4

x

y

1

1

f

x

y

1

1

f

12. Geometry The volume of a cube with side length x inches is given by V1 5 x3.

The volume of a cube with side length (x 1 1) inches is given by V2 5 (x 1 1)3.

a. Copy and complete the table.

x 1 2 3 4 5 6

V1

V2

b. Use the table from part (a) to graph V1 and V2. Explain how the graphs are related.

0 2 41 3 5 6 x0

50

100

150

200250

300

350V

Side length (inches)

Vo

lum

e (

cu

bic

in

ch

es)

c. Find the volume of each cube when x 5 8.



2.3 Factor and Solve Polynomial EquationsGoal p Factor and solve polynomial equations.Georgia


MM3A3b, MM3A3d

Your Notes

VOCABULARY

Factored completely

Factor by grouping

Quadratic form

SPECIAL FACTORING PATTERNS

Sum of Two Cubes

a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)Example

x3 1 8 5 (x 1 2)( )Difference of Two Cubes

a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2)Example

8x3 2 1 5 (2x 2 1)( )

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Your Notes

Factor the polynomial completely.

a. z3 2 125 5 z3 2 Difference of two cubes

5 (z 2 )( )

b. 81y4 1 192y 5 3y( ) Factor common monomial.

5 3y[ 1 ] Sum of two cubes

5 3y( )( )

Example 1 Factor the sum or difference of two cubes

1. Factor the polynomial 8x3 1 64 completely.


Factor the polynomial x3 2 2x2 2 9x 1 18 completely.

x3 2 2x2 2 9x 1 18 5 x2( ) 2 9( ) Factor by grouping.

5 Distributive property

5 Difference of two squares

Example 2 Factor by grouping

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Your Notes

Factor completely: (a) 16x4 2 256 and (b) 3y7 2 15y5 1 18y3.

a. 16x4 2 256 5 ( )2 2 2

5

5

b. 3y7 2 15y5 1 18y3 5 3y3( )

5

Example 3 Factor polynomials in quadratic form

2. x3 1 2x2 2 25x 2 50 3. x4 2 14x2 1 45

Checkpoint Factor the polynomial completely.

Find the real-number solutions of the equation x4 1 9 5 10x2.

x4 1 9 5 10x2 Write original equation.

5 0 Write in standard form.

5 0 Factor trinomial.

5 0 Difference of two squares

x 5 , x 5 , x 5 , x 5 Zero product property

The solutions are .

Example 4 Find real-number solutions

4. 2x5 1 24x 5 14x3

Checkpoint Find the real-number solutions.Homework

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Name ——————————————————————— Date ————————————

LESSON

2.3 PracticeFind the greatest common factor of the terms in the polynomial.

1. 4x4 1 12x3 2. 10y2 1 4y 2 64 3. 16x5 2 8x

4. 32n5 2 64n3 1 16n2 5. 15p6 2 5p4 2 10p2 6. 36c9 1 13

Match the polynomial with its factorization.

7. 3x2 1 11x 1 6 A. 2x3(x 1 2)(x 2 2)(x2 1 3)

8. x3 2 4x2 1 4x 2 16 B. 2x(x 1 4)(x 2 4)

9. 125x3 2 216 C. (3x 1 2)(x 1 3)

10. 2x7 2 2x5 2 24x3 D. (x2 1 4)(x 2 4)

11. 2x5 1 4x4 2 4x3 2 8x2 E. 2x2(x2 2 2)(x 1 2)

12. 2x3 2 32x F. (5x 2 6)(25x2 1 30x 1 36)

Factor the sum or difference of cubes.

13. s3 2 1 14. q3 1 1

15. x3 2 27 16. a3 1 125

17. h3 1 64 18. 8y3 2 125



Factor the polynomial by grouping.

19. x3 1 2x2 1 3x 1 6 20. z3 2 z2 1 5z 2 5

21. f 3 1 4f 2 1 f 1 4 22. m3 2 2m2 1 4m 2 8

23. 2x4 2 x3 1 6x 2 3 24. t3 2 2t2 2 9t 1 18

Find the real-number solutions of the equation.

25. w2 2 3w 5 0 26. v3 1 5v2 5 0

27. x2 2 5x 1 6 5 0 28. d2 2 16 5 0

29. 10s3 5 30s2 30. x3 1 x2 2 9x 2 9 5 0

Match the equation for volume with the appropriate solid.

31. V 5 x3 2 4x 32. V 5 x3 2 4x2 1 4x 33. V 5 x4 2 16

A. x 2 2

x 2 2x

B.

x 2 2x 1 2

x2 1 4

C.

x 1 2

x 2 2x

LESSON


Name ——————————————————————— Date ————————————



2.4 Solve Polynomial InequalitiesGoal p Solve polynomial inequalities.Georgia


MM3A3c

Your Notes

VOCABULARY

Polynomial inequality

Intervals

INEQUALITY AND INTERVAL NOTATION

In the intervals below, the real numbers a and b are the endpoints of each interval.

Bounded intervals:

Inequality a ≤ x ≤ b a < x < b a ≤ x < b a < x ≤ b

Notation

Unbounded intervals:

Inequality x ≥ a x > a x ≤ b x < b

Notation

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Your Notes

Solve x3 2 x2 < 12x algebraically.

First, write and solve the equation obtained by replacing with .

x3 2 x2 5 12x Write equation that corresponds to original inequality.

x3 2 x2 2 5 0 Write in standard form.

5 0 Factor.

x 5 , x 5 , or x 5 Zero product property

The numbers 0, 4, and 23 are the of the inequality x3 2 x2 < 12x. Plot 0, 4, and 23 on a number line, using because the values do not satisfy the inequality. The critical x-values partition the number line into four intervals. Test an x-value in each interval to see if it satisfies the inequality.

2124 2223 0 1 2 3 4 5 6 7 8

Test x 5 5:53 2 52 5 ,

Test x 5 21:(21)3 2 (21)2 5 ,

Test x 5 1:13 2 12 5 ,

Test x 5 24:(24)3 2 (24)2 5 ,

The solution set consists of all real numbers in the intervals and .

Example 1 Solve a polynomial inequality algebraically

1. Solve x3 2 16x > 0 algebraically.


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Your Notes

Solve 2x3 1 7x2 2 4x ≥ 0 by graphing.

The solution consists of the x-values for which the graph of y 5 2x3 1 7x2 2 4x lies or the x-axis.Find the graph's x-intercepts by letting y 5 0 and solve for x.

2x3 1 7x2 2 4x 5 0 Set y equal to 0.

5 0 Factor.

x 5 , x 5 , or x 5 Zero product property

Graph the polynomial and plot the

x

y

10

1

the x-intercepts , , and .

The graph lies on or above the x-axisbetween (and including) x 5and x 5 and to the right of

(and including) x 5 . The solution set consists of all

real numbers in the intervals and .

Example 2 Solve a polynomial inequality by graphing

2. Solve x4 2 4x2 ≤ 0 by graphing.


Homework

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Represent the inequality using interval notation.

1. x > 8 2. x ≤ 0 3. 4 ≤ x < 6

Represent the inequality using inequality notation.

4. [22, 3] 5. (21, 1) 6. [5, 1`)

Tell whether the given x-value is a solution of the inequality.

7. x3 2 4x < 0; x 5 1 8. x3 2 5x2 1 4x ≤ 0; x 5 1 } 2 9. x4 2 8 > 0; x 5 5

10. x3 2 x2 ≥ 6x; x 5 23 11. x3 1 4x2 ≥ 23x; x 5 4 12. x4 2 3x2 < 22; x 5 0

Solve the inequality algebraically.

13. x3 2 1 ≥ 0 14. x3 1 x2 2 6x ≤ 0 15. x4 2 1 < 0

16. x3 1 3x2 > 10x 17. 3x3 1 x2 ≥ 2x 18. x4 2 10x2 < 29

LESSON

2.4 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Use the graph of the corresponding equation to determine the solution set of the inequality.

19. x3 2 x2 2 2x ≤ 0 20. x3 2 2x2 2 8x > 0 21. x4 2 3x2 2 4 < 0

x

y

1

1

y 5 x3 2 x2 2 2x

x

y

5

1

y 5 x3 2 2x2 2 8x

x

y

5

1

y 5 x4 2 3x2 2 4

Solve the inequality using a graph.

22. x3 2 4x2 1 4x ≤ 0 23. 2x3 1 3x2 2 2x > 0 24. x4 2 8 ≥ 0

x

y

1

1

x

y

1

1

x

y

2

1

25. Geometry The volume V of a cube with a side length of x inches is greater than 27 cubic inches.

a. Copy and complete the table to determine the possible side lengths of the cube.

x 1 2 3 4 5 6

V 5 x3

b. Check the solution you found in part (a) by solving the inequality x3 > 27 algebraically.



2.5 Apply the Remainder and Factor TheoremsGoal p Use theorems to factor polynomials.Georgia


MM3A3a

Your Notes

VOCABULARY

Polynomial long division

Synthetic division

Divide f(x) 5 4x4 1 5x2 2 9x 1 18 by x2 1 2x 1 4.

Write polynomial division in the same format you use when dividing numbers. Include a “0” as the coefficient of x3. At each stage, divide the term with the highest power in what is left of the dividend by the first term of the divisor. This gives the next term of the quotient.

x2 1 2x 1 4qwww4x4 1 0x3 1 5x2 2 9x 1 18

Write the result:

4x4 1 5x2 2 9x 1 18}}

x2 1 2x 1 45

Example 1 Use polynomial long division

You can check the result of a division problem by multiplying the quotient by the divisor and adding the remainder. The result should be the dividend.

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Your NotesREMAINDER THEOREM

If a polynomial f (x) is divided by x 2 k, then the remainder is r 5 .

Divide f(x) 5 x3 1 4x2 2 5x 1 3 by x 1 2.

Solution22 1 4 25 3

x3 1 4x2 2 5x 1 3}}

x 1 2 5

Example 2 Use synthetic division

FACTOR THEOREM

A polynomial f (x) has a factor x 2 k, if and only if f (k) 5 .

Factor f(x) 5 2x3 2 11x2 1 3x 1 36 completely given that x 2 3 is a factor.

Solution

Because x 2 3 is a factor of f(x), you know that f(3) 5 .Use synthetic division to find the other factors.

2 211 3 36

Use the result to write f(x) as a product of two factors and then factor completely.

f (x) 5 2x3 2 11x 1 3x 1 36 5 ( )( ) 5 ( )( )( )

Example 3 Factor a polynomial

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Your Notes

Find the other zeros of f(x) 5 x3 1 4x2 2 15x 2 18 given that f(21) 5 0.

Solution

Because f (21) 5 0, is a factor of f. Use synthetic division to find the other factors.

1 4 215 218

Use the result to write f(x) as a product of two factors and then factor completely.

f (x) 5 x3 1 4x2 2 15x 2 18

5 ( )( )

5 ( )( )( )

The zeros are .

Example 4 Finding zeros of functions

1. Use long division to divide x3 2 6x2 1 9 by x 2 4.

2. Use synthetic division to divide 2x3 1 4x2 2 3x 2 6 by x 1 3.

3. Factor f (x) 5 3x3 1 8x2 1 3x 2 2 given that x 1 2 is a factor.

4. Find the other zeros of f (x) 5 x3 2 4x2 2 11x 1 30 given that f(2) 5 0.

Checkpoint Complete the following exercises.

Homework

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Name ——————————————————————— Date ————————————

LESSON

2.5 PracticeWrite the divisor, dividend, quotient, and remainder represented by the synthetic division.

1. 22 2 1 21 10

24 6 210

2 23 5 0

2. 4 3 210 0 25

12 8 32

3 2 8 27

Divide using polynomial long division.

3. (x2 2 6x 1 10) 4 (x 2 1) 4. (x2 1 2x 2 11) 4 (x 1 2)

5. (x2 1 3x 2 18) 4 (x 1 6) 6. (x2 1 3x 2 6) 4 (x 1 5)

7. (4x2 2 7x 2 4) 4 (x 2 4) 8. (2x2 2 x 1 5) 4 (x 1 3)

9. (x2 1 4) 4 (x 2 2) 10. (x3 1 11x2 1 25x 2 21) 4 (x 1 7)



Divide using synthetic division.

11. (x2 1 8x 2 9) 4 (x 1 9) 12. (x2 2 x 2 1) 4 (x 1 1)

13. (x2 1 3x 2 10) 4 (x 2 2) 14. (x2 2 6x 1 4) 4 (x 1 3)

15. (x2 1 5x 2 7) 4 (x 1 4) 16. (2x2 2 7x 2 15) 4 (x 2 5)

17. (x3 1 x 1 2) 4 (x 2 1) 18. (x2 2 7) 4 (x 1 2)

You are given an expression for the area of the rectangle. Find an expression for the missing dimension.

19. A 5 x2 1 10x 1 21 20. A 5 x2 1 2x 2 8 21. A 5 x2 1 8x 1 15

x 1 3

?

x 1 4

?

x 1 5

?

22. Publishing The profi t P (in thousands of dollars) for an educational publisher can be modeled by P 5 2b3 1 5b2 1 b where b is the number of workbooks printed (in thousands). Currently, the publisher prints 5000 workbooks and makes a profi t of $5000. What lesser number of workbooks could the publisher print and still yield the same profi t?

LESSON


Name ——————————————————————— Date ————————————



2.6 Find Rational ZerosGoal p Find all real zeros of a polynomial function.Georgia


MM3A3a, MM3A3d

Your Notes

THE RATIONAL ROOT THEOREM

If f(x) 5 an x n 1 . . . 1 a1x 1 a0 has coefficients, then every rational zero of f has the following form:

p}q 5

factor of constant term }}}factor of leading coefficient

Find all real zeros of f(x) 5 x3 2 4x2 2 7x 1 10.

1. List the possible rational zeros. The leading coefficient is and the constant term is . So, the possible

rational zeros are: x 5 , , ,

2. Test these zeros using synthetic division. Test x 5 :

1 24 27 10

is a zero.

3. Factor the trinomial and use the factor theorem.

f (x) 5 ( )( )

5

The zeros of f are .

Example 1 Find zeros when the leading coefficient is 1

1. f (x) 5 x3 1 3x2 2 10x 2 24

Checkpoint Find all real zeros of the function.

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Your Notes Example 2 Find zeros when the leading coefficient is not 1

Find all real zeros of f(x) 5 8x4 1 2x3 2 21x2 2 7x 1 3.

1. List the possible rational zeros of f:

2. Choose a reasonable value to check using the graph of the function.

3. Check x 5 :

8 2 221 27 3

is a zero. 4. Factor out a binomial.

f (x) 5 Write as a product of factors.

5 Factor out .

5 Multiplyby .

5. Repeat the steps above for g(x) 5 .Any zero of g will also be a zero of f. Synthetic division

shows that is a zero and yields the quotient

. Factoring a 4 out of the quotient yields f (x) 5 .

6. Find the remaining zeros by solving 5 0.

x 5 Use quadratic formula.

x 5 Simplify.

The real zeros of f are .

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Your Notes

Sandbox You are building a wooden square sandbox for a local playground. You want the volume of the box to be 16 cubic feet. You want the height of the box to be x feet and the length of each side of the square base to be x 1 3 feet. What are the dimensions?

1. Write an equation for the volume of the sandbox. The volume is V 5 Bh where B 5 base area and h 5 height.

Volume (cubic feet)

5Area of base (square feet)

pHeight(feet)

16 5 (x 1 3)2 p x

16 5 Write the equation.

16 5 Multiply.

0 5Subtract from each side.

2. List the possible rational solutions:

3. Test the possible rational solutions. Only positive x-values make sense.

1

4. Check for other solutions. The other possible rational solutions solutions, so x 5 is the solution. The height of the sandbox should be

foot and each side of the base should be 5 feet.

Example 3 Solve a multi-step problem

2. f (x) 5 9x4 1 12x3 2 26x2 2 11x 1 6

Checkpoint Find all real zeros of the function.Homework

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1. Can you use the rational root theorem to fi nd the zeros of the polynomial function f (x) 5 0.4x2 2 3x 1 2.2? Explain why or why not.

List the possible rational zeros of the function using the rational root theorem.

2. f (x) 5 x3 2 5x 1 16 3. g(x) 5 x4 1 8x2 2 18

4. h(x) 5 x5 1 2x4 2 3x 2 24 5. f (x) 5 x8 2 2x5 1 x4 2 3x 1 20

6. h(x) 5 2x3 2 5x2 2 9 7. g(x) 5 3x3 1 7x 1 12

Use synthetic division to decide which of the following are zeros of the function: 23, 21, 1, 3.

8. f (x) 5 x3 2 9x 1 3 9. g(x) 5 x4 1 3x3 2 7x2 2 27x 2 18

10. g(x) 5 2x4 2 9x3 1 8x2 1 9x 2 10 11. h(x) 5 3x4 1 5x2 2 17x 1 9

Find all rational zeros of the function.

12. f (x) 5 x3 1 x2 2 14x 2 24 13. f (x) 5 x3 2 2x2 2 x 1 2

x

y

10

2

x

y

1

1

LESSON

2.6 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Find all real zeros of the function.

14. f (x) 5 x3 2 8x2 2 23x 1 30 15. g(x) 5 x3 1 2x2 2 11x 2 12

16. h(x) 5 x3 2 7x2 1 2x 1 40 17. h(x) 5 x3 1 9x2 2 4x 2 36

18. g(x) 5 x4 2 5x3 1 7x2 1 3x 2 10 19. f (x) 5 x4 2 2x3 2 7x2 1 8x 1 12

20. f (x) 5 x4 1 3x3 2 21x2 2 43x 1 60 21. g(x) 5 x4 1 x3 2 11x2 2 9x 1 18

22. Crafts You have 18 cubic inches of wax, and you

x

x 1 3

want to make a candle in the shape of a pyramid with a square base as shown.

a. Write an equation that shows that the volume of the candle is 18 cubic inches.

b. Use the rational root theorem to list all possible rational solutions of the equation in part (a).

c. Find all real values of x that are valid as a dimension of the candle.

d. Find the dimensions of the candle.



2.7 Apply the Fundamental Theorem of AlgebraGoal p Classify the zeros of polynomial functions.Georgia


MM3A3a

Your Notes

VOCABULARY

Repeated solution

THE FUNDAMENTAL THEOREM OF ALGEBRA

Theorem: If f (x) is a polynomial of degree n where n 0, then the equation f (x) 5 0 has at least solution in the set of complex numbers.

Corollary: If f (x) is a polynomial of degree n wheren 0, then the equation f (x) 5 0 has exactly

solutions provided each solution repeated twice is counted as solutions, each solution repeated three times is counted as solutions, and so on.

Find the number of solutions or zeros for each equation or function.

a. Because x3 2 3x2 1 9x 2 27 5 0 is a degree polynomial equation, it has solutions.

b. Because f (x) 5 x4 1 6x3 2 32x is a degree polynomial function, it has zeros.

Example 1 Find the number of solutions or zeros

1. State the number of zeros of

f (x) 5 x3 2 2x2 2 9x 1 18.


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Your Notes

Find all zeros of f(x) 5 x5 2 5x4 2 9x3 2 5x2 2 8x 1 12.

Solution1. Find the rational zeros of f. Because f is a fifth-degree

function, it has zeros. The possible rational zeros are . Using synthetic division, you can determine that is a zero repeated twice and is also a zero.

2. Write f (x) in factored form. Dividing f by its known factors gives a quotient of . So,

f (x) 5 .

3. Find the complex zeros of f. Use the quadratic formula to factor the trinomial into linear factors.

f (x) 5

The zeros of f are .

Example 2 Find the zeros of a polynomial function

2. f (x) 5 x4 2 7x3 1 13x2 1 x 2 20

Checkpoint Find all zeros of the polynomial function.

COMPLEX CONJUGATES THEOREM

If f is a polynomial function with coefficients, and is an imaginary zero of f, then is also a zero of f.

IRRATIONAL CONJUGATES THEOREM

Suppose f is a polynomial function with coefficients, and a and b are rational numbers such that Ï

}

b is irrational. If is a zero of f, then is also a zero of f.

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Your Notes

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 22 and 3 1 i as zeros.

Because the coefficients are real and 3 1 i is a zero, must also be a zero. Use the three zeros and the

factor theorem to write f (x) as a product of three factors.

f (x) 5 ( )[x 2 ( )][x 2 ( )] Factored form

5 ( )[ ][ ] Regroup terms.

5 Multiply.

5 Expand, usei2 5 21.

5 Simplify.

5 Multiply.

5 Combinelike terms.

Example 3 Use zeros to write a polynomial function

3. Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 4 and 1 1 Ï

}

6 as zeros.


DESCARTES’ RULE OF SIGNS

Let f (x) 5 anxn 1 an 2 1xn 2 1 1 . . . 1 a2x2 1 a1x 1 a0be a polynomial function with real coefficients.

• The number of real zeros of f is equal to the number of changes in sign of the coefficients of or is less than this by an number.

• The number of real zeros of f is equal to the number of changes in sign of the coefficients of or is less than this by an number.

You can check this result by evaluating f (x) at each of its three zeros.

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Your Notes

Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for

f(x) 5 2x5 2 7x4 1 12x3 1 2x2 1 4x 1 6.

Solutionf(x) 5 2x5 2 7x4 1 12x3 1 2x2 1 4x 1 6

The coefficients of f (x) have sign changes, so f has positive real zero(s).

f (2x) 5 2(2x)5 2 7(2x)4 1 12(2x)3

1 2(2x)2 1 4(2x) 1 6

5

The coefficients of f (2x) have sign changes, so f has negative real zero(s).

Positive real zeros

Negative real zeros

Imaginary zeros

Total zeros

Example 4 Use Descartes’ rule of signs

4. Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f (x) 5 3x5 2 4x4 1 x3 1 6x2 1 7x 2 8.


Homework

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Identify the number of solutions or zeros.

1. 2x2 1 5x 2 9 5 0 2. f (t) 5 t3 1 4t2 2 7

3. g(z) 5 2z3 1 z2 1 6z 2 3 4. x4 2 6x2 1 8x 2 12 5 0

5. h(x) 5 2x5 2 3x4 1 x 2 9 6. 26x4 1 7x2 2 15 5 0

7. f (y) 5 22y 1 3 8. 3r3 2 r2 1 5r 2 21 5 0

Given that f (x) has real coeffi cients and x 5 k is a zero, what other number must be a zero?

9. k 5 3 2 Ï}

2 10. k 5 i 11. k 5 22 2 9i

12. k 5 Ï}

3 2 i 13. k 5 1 1 Ï}

5 i 14. k 5 Ï}

2 1 Ï}

7 i

Find all the zeros of the polynomial function.

15. g(x) 5 x3 2 8x2 2 15x 1 54 16. f (x) 5 x4 2 3x3 2 17x2 1 39x 2 20

17. h(x) 5 x3 1 6x2 1 x 1 6 18. g(x) 5 x3 1 8x2 2 7x 2 56

19. f (x) 5 x3 2 2x2 1 9x 2 18 20. h(x) 5 x4 2 x2 2 20

LESSON

2.7 Practice

Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


Write a polynomial function f of least degree that has rational coeffi cients, a leading coeffi cient of 1, and the given zeros.

21. 29 22. 25, 4

23. 23, 21 24. 21, 0, 1

25. 21, 2, 6 26. 22, 21, 10

27. The graph f (x) 5 x3 2 x2 2 8x 1 12 is shown at the

x

y

1

1

right. How many real zeros does the function have? How many imaginary zeros does the function have?

28. Geometry A square piece of sheet metal is

10 in.x in.

10 in.

x in.

10 inches by 10 inches. Squares of side length x are cut from the corners and the remaining piece is folded to make an open box. The volume of the box is modeled by V(x) 5 4x3 2 40x2 1 100x. What size square(s) can be cut from the corners to give a box with a volume of 25 cubic inches?



2.8 Analyze Graphs of Polynomial FunctionsGoal p Use intercepts to graph polynomial functions.Georgia


MM3A1b, MM3A1d

Your Notes

VOCABULARY

Local maximum

Local minimum

Multiplicity of a root

MULTIPLICITY OF A ROOT

For the polynomial equation f(x) 5 0, k is a repeated solution, or a root with a ,if and only if the factor x 2 k has an exponent greater than when f(x) is factored completely. If the exponent is , the graph of f the x-axisat the zero. If the exponent is , the graph of f the x-axis at the zero.

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Your Notes

1. Graph the function y

x

y

1

1

f (x) 5 2(x 2 2)(x 1 1)(x 2 1).


TURNING POINTS OF POLYNOMIAL FUNCTIONS

The graph of every polynomial function of degree n has at most turning points.

Graph the function y

x

y

1

1

f(x) 51}4

(x 1 1)2(x 2 4).

1. Use the intercepts. Because and are zeros of f,

plot ( , ) and ( , ).

2. Plot points between and beyond the x-intercepts.

x 22 0 1 2 3

y

3. Determine the end behavior and multiplicity. Because f has factors of the form x 2 k, and a

constant factor of , it is a function with

a leading coefficient. So, f (x) → asx → 2` and f (x) → as x → 1`. The zero 21 is repeated . So, the graph of f the x-axis at (21, 0).

4. Draw the graph so that it passes through the plotted points and has the appropriate end behavior.

Example 1 Use x-intercepts to graph a polynomial function

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Your Notes

2. Use a graphing calculator to identify the x-intercepts, local maximums, and local minimums of the graph of f (x) 5 x4 1 x3 2 5x2 1 4.


Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur.

a. f (x) 5 x3 2 4x2 1 6

b. f (x) 5 2x4 1 3x3 1 x2 2 4x

a. Use a graphing calculator to graph the function.Notice that the graph of f has

x-intercepts and turning points. Use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points.

The x-intercepts of the graph are . The function has a local maximum at

( , ) and a local minimum at ( , ).

b. Use a graphing calculator to graph the function. Notice that the graph of f has

x-intercepts and turning points. Use the graphing calculator’s zero, maximum, and minimum features to approximate the coordinates of the points.

The x-intercepts of the graph are . The function

has local maximums at ( , ) and ( , ). The function has a local minimum at ( , ).

Example 2 Find turning points

Homework

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Name ——————————————————————— Date ————————————

LESSON

2.8 Practice 1. True or False If k is a zero of the polynomial function f, then k is an x-intercept of

the graph of f (x). Explain your answer.

Determine the lowest-degree polynomial that has the given graph.

2. y

x2

2

3. y

x1

1

4. y

x2

2

Estimate the coordinates of each turning point and state whether each corresponds to a local maximum or a local minimum.

5. y

x2

2

6. y

x1

1

7. y

x1

1

Match the graph with its function.

8. f (x) 5 2x4 2 3x2 2 2 9. f(x) 5 2x6 2 6x4 1 4x2 2 2 10. f (x) 5 22x4 1 3x2 2 2

A.

x

y

1

1

B.

x

y

1

1

C.

x

y

1

1



Determine the x-intercepts of the function.

11. g(x) 5 (x 1 4)(x 2 1) 12. h(x) 5 (x 2 2)(x 2 3)

13. f (x) 5 x(x 1 4)(x 2 5) 14. f (x) 5 (x 1 3)(x 1 1)(x 2 8)

15. g(x) 5 (x 1 6)2 16. h(x) 5 (x 2 1)(x 2 7)2

Graph the function.

17. f (x) 5 (x 1 1)(x 2 2) 18. g(x) 5 (x 2 3)(x 2 1) 19. h(x) 5 (x 1 6)(x 1 7)

x

y

1

1

x

y

1

1

x

y

2

22

20. h(x) 5 0.9(x 1 5)(x 2 2) 21. g(x) 5 (x 2 3)2 22. f (x) 5 (x 1 1)(x 2 1)(x 2 3)

x

y

222

x

y

1

1

x

y

1

1

LESSON


Name ——————————————————————— Date ————————————



Name ——————————————————————— Date ————————————

LESSON


23. Let f be a fourth-degree polynomial function with the zeros 22, 6, 2i, and 22i.

a. How many distinct linear factors does f (x) have?

b. How many distinct solutions does f (x) 5 0 have?

c. What are the x-intercepts of the graph of f ?

24. Manufacturing You are designing an open box from a piece of cardboard that is 18 inches by 18 inches. Squares of side length x are cut from the corners and the remaining piece is folded to make an open box. The volume of the box is given by the function

V 5 4x3 2 72x2 1 324x.

Using a graphing calculator, you would obtain the graph shown below.

a. What is the domain of the volume function? Explain.

b. Use the graph to estimate the length of the cut x that will maximize the volume of the box.

c. Estimate the maximum volume the box can have.



Words to ReviewGive an example of the vocabulary word.

Polynomial

Degree of a polynomial function

Standard form of a polynomial function

End behavior

Odd function

Polynomial function

Leading coefficient

Synthetic substitution

Even function

Factored completely

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Factor by grouping

Polynomial inequality

Polynomial long division

Quadratic form

Intervals

Synthetic division

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Repeated solution

Local minimum

Local maximum

Multiplicity of a root

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evaluate and graph polynomial functions -...

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