1

In Problems 1 – 4, determine whether the function is a polynomial function, rational function, or neither.

For those that are polynomial functions, state the degree. For those that are not polynomial functions, tell

why not.

1. f(x) = -x4 – 3x3 + 1

4 3( ) 3 1= − − +f x x x is a polynomial of degree 4.

2. 2 1( ) 2 4 1G x x x− −= + −

2 12

2 4( ) 2 4 1 1− −= + − = + −f x x x

xx is a rational

function. It is not a polynomial function since the

variable x is raised to a negative power.

3. 2

4( )

7 3

xR x

x x

−=

− +

2

4( )

7 3

−=

− +

xf x

x x is not a polynomial function. It

is the ratio of two polynomials so it is a rational

function.

4. H(x) = -4

( ) 4=−f x is a polynomial function of degree 0. It

is also a rational function since it can be expressed

as 4

( )1

−=f x .

In Problems 5 – 7, graph each function using transformations (shifting, compressing, stretching, and

reflections). Show all the stages.

5. f(x) = -x3 + 3

Using the graph of 3=y x , reflect

about the x-axis, then shift up 3

units.

6. f(x) = (x – 1)4 – 2

Using the graph of 4=y x , shift

right 1 unit, then shift down 2

units.

7. f(x) = (1 – x)3

( ) ( )33

( ) 1 1= − = − − f x x x

Using the graph of 3=y x , shift

right 1 unit, then reflect about the

x-axis.

2

In Problems 8 – 11, analyze each polynomial by following Steps 1 through 8.

8. f(x) = x(x – 2)(x – 4)

Step 1: Degree is 3. The function resembles 3y x=

for large values of x .

Step 2: y-intercept: (0) 0(0 2)(0 4) 0f = − − =

x-intercepts: solve ( ) 0f x =

( 2)( 4) 0

0 or 2 or 4

x x x

x x x

− − =

= = =

Step 3: Real zeros: 0 with multiplicity one, 2

with multiplicity one, 4 with multiplicity one. The

graph crosses the x-axis at x = 0, x = 2, and x = 4.

Step 4: Graphing utility:

Step 5: 2 turning points;

local maximum: (0.85, 3.08) ;

local minimum: (3.15, 3.08)−

Step 6: Graphing by hand:

Step 7: Domain: ( ),− ; Range: ( ),−

Step 8: Increasing on ( , 0.85)− and (3.15, ) ;

decreasing on (0.85, 3.15)

9. f(x) = (x – 2)(x + 4)2

Step 1: Degree is 3. The function resembles 3y x=

for large values of x .

Step 2: y-intercept: 2(0) (0 2)(0 4) 32f = − + = −

x-intercepts: solve ( ) 0f x = 2( 2)( 4) 0

2 or 4

x x

x x

− + =

= = −

Step 3: Real zeros: 4− with multiplicity two,

2 with multiplicity one. The graph touches the x-axis

at 4x = − and crosses it at 2x = .

Step 4: Graphing utility:

Step 5: 2 turning points; local maximum: ( 4, 0)− ;

local minimum: (0, 32)−

Step 6: Graphing by hand:

Step 7: Domain: ( ),− ; Range: ( ),−

Step 8: Increasing on ( , 4)− − and (0, ) ;

decreasing on ( 4, 0)−

3

10. 3( ) 4 4f x x x= − +

3 2( ) 4 4 4 ( 1) 4 ( 1)( 1)f x x x x x x x x= − + = − − = − − +

11. f(x) = (x – 4)(x + 2)2(x – 2)

4

In Problems 12 and 13, find the remainder R when f(x) is divided by g(x). Is g a factor of f?

12. f(x) = 2x3 + 8x2 – 5x + 5; g(x) = x – 2

3 2(2) 2(2) 8(2) 5(2) 5

2(8) 8(4) 10 5

43

f = + − +

= + − +

=

So 43R = and g is not a factor of f .

13. f(x) = x4 – x2 + 2x + 2; g(x) = x + 1

4 2( 1) ( 1) ( 1) 2( 1) 2 1 1 2 2 0f − = − − − + − + = − − + =

So 0R = and g is a factor of f .

14. Find the value of f(x) = -16x3 + 18x2 – x + 2 at x = -2.

2 16 18 1 2

32 100 202

16 50 101 204

− − −

−

− −

( 2) 204− =f

15. List all the potential rational zeros of f(x) = -6x5 + x4 + 2x3 – x + 1.

The maximum number of zeros is the degree of the polynomial, which is 5. p must be a factor of 1: 1=p

q must be a factor of 6− : 1, 2, 3, 6= q

The possible rational zeros are: 1 1 1

1, , ,2 3 6

= p

q

5

In Problems 16 – 18, use the Rational Zeros Theorem to find all the real zeros of each polynomial function.

Use the zeros to factor f over the real numbers.

16. f(x) = x3 – x2 – 10x – 8

17. f(x) = 4x3 – 4x2 – 7x – 2

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18. f(x) = x4 + 6x3 + 11x2 + 12x + 18

In Problems 19 and 20, solve each equation in the real number system.

19. 3x4 + 3x3 – 17x2 + x – 6 = 0

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20. 2x4 + 7x3 – 5x2 – 28x – 12 = 0

In Problems 21 and 22, information is given about a complex polynomial f(x) whose coefficients are real

numbers. Find the remaining zeros of f. Then find a polynomial function with real coefficients that has the

zeros.

21. Degree: 3; zeros: 3 + 4i, 5

22. Degree 4; zeros: 1, 2, 1 + i

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In Problems 23 – 26, find the complex zeros of each polynomial function f(x). Write f in factored form.

23. f(x) = x3 – x2 – 10x – 8 (See Problem 16)

24. f(x) = 4x3 – 4x2 – 7x – 2 (See Problem 17)

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25. f(x) = x4 + 6x3 + 11x2 + 12x + 18

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26. f(x) = 3x4 + 3x3 – 17x2 + x – 6

In Problems 27 and 28, find the domain of each rational function. Find any horizontal, vertical, or oblique

asymptotes.

27. 2 4

( )2

xR x

x

+=

− 28.

2

2

2 1( )

5 6

xR x

x x

+=

− +

11

In Problems 29 – 34, discuss each rational function following Steps 1 – 7.

29. 4

( )x

R xx

−=

12

30. 2

( )1

xR x

x=

−

13

31. 2

2

6 9( )

x xR x

x

− +=

32. 3

2( )

2 1

xR x

x x=

− +

14

33. 2

2

2( )

9

xR x

x=

−

15

34. 2

2

2 1( )

1

x xR x

x

− +=

−

In Problems 35 –38, solve each inequality. Graph the solution set.

35. x3 + x2 < 4x + 4

36. 6

13x

+

16

37. ( 2)( 1)

03

x x

x

− −

−

38. 2

2

8 120

16

− +

−

x x

x

39. Write a polynomial function of degree 4 with the following zeros: 3 (multiplicity 2); -1 (multiplicity

1); -3 (multiplicity 1), and y-intercept -157.

23 1 3 184( ) ( ) ( )( )f x x x x= − + + −