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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
6
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
7
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
8
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)
9
25. f(x) = x4 + 6x3 + 11x2 + 12x + 18
10
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= − + + −
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