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Page 1: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 1

Page 2: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 2

Polynomial and Rational Functions

Polynomial Division Questions

1) Use long division to perform the following division problems:

a. Divide 3 24 5 11x x x+ − − by 24 6x x+ − .

b. Divide 4 3 23 10 17 10 24x x x x+ + + + by 23 14 8x x+ + .

2) Use long division to evaluate the following problems:

a. 5 4 3 2

2

8 15 20 76 48

6

x x x x x

x x

− + + − +

− −

b. 5 4 3 2

3 2

8 15 20 76 48

2 5 6

x x x x x

x x x

− + + − +

− − +

3) Use long division to solve the following division problems:

a. Divide 3 23 3 3 1x x x+ + + by 6x+ .

b. Divide 3 212 10 8x x x+ + + by 24 3 1x x+ + .

4) Use long division to solvethe following division problems:

a. Divide 4 3 24 3 4 3x x x x+ + − − by 2 10 9x x+ − .

b. Divide 4 3 23 12 8 10 2x x x x− + − + by 3 2 7 1x x x+ + + .

5) Use long division to solve the following division problems:

a. Divide 3 2 10 8x x x+ − + by 2 3 4x x+ −

b. Divide 4 3 22 14 19 6x x x x− − + − by 22 3 1x x− +

Page 3: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 3

6) Use long division to evaluate the following:

a. ( )( )

6 5 3 4 22 13 31 31 8 4 7

2 1 2

− − + + − +

+ −

x x x x x x

x x

b. ( )

7 4 3 2

2

2 25 120 270 270 81

1

+ + + + +

x x x x x

x

7) Use long division to evaluate the following:

a. 4 3x ax x a

x a

− + −

b. ( ) ( )3 2

2

1 2 2

2

x b x b x b

x x

+ − − + +

− +

Answer Key

1) a. 2

4 51

4 6

xx

x x

−+ +

+ − b. 2

2

4 7108 49

4 2 9 993 9 3 14 8

x

x xx x

+

− + −+ +

2) a. 3 27 14 8x x x− + − b. 2 6 8x x− +

3) a. 2 5573 15 93

6x x

x− + −

+ b.

( )2

2

21 834 3 1

16 4 3 1

xx x

x x

++ + +

+ +

4) a. 2

2

778 6456 72

10 9

xx x

x x

− +− + +

+ − b.

2

3 2

2 92 173 15

7 1

x xx

x x x

+ +− +

+ − +

5) a. 2x− b. 2 6x x+ −

6) a. 4 3 25 9 7 2x x x x− + − + b. 5 4 3 22 4 6 33 80 43x x x x x+ + + + −

7) a. 3 1x + b. x b−

Page 4: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 4

Rationals

Questions

1) In sections a-c, reduce the rational expression to the lowest term possible:

a. 2

2

4

2 5

x x

x x

+

− b.

2

1

2 3

x

x x

+ − c.

2

2

2 8

4

x x

x

− −

2) In sections a–d, perform the indicated operation, and reduce the rational expression to the lowest term possible:

a. 2 2

2 2

2 2 15

6 3 4

x x x x

x x x x

+ − + −

− − + − b.

2 2

2 2

9 2 3:

4 21 9 14

x x x

x x x x

− + −

+ − + +

c. 2 2

2 2

4 3 6 27:

4 5 14 45

x x x x

x x x x

− + + −

+ − + + d.

2

2

3

45

8 20

xx

x x

−+

+ −

3) In sections a–f, perform the indicated operations:

a. 2

1 3 5

4 4 4x x x+ −

− + − b.

2

3 1 3

1x x x+ +

+

c. 2

1

11 30 5

x x

x x x

−+

+ + + d.

2

2

1 1

6 7 7 1

x x

x x x x

++ +

− − − +

e. 3 5 2

4 1x x x+ −

− +

f. ( ) ( )

3 2

1

2 3 2 3

x x

x x

++

+ +

Page 5: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 5

Answer Key

1) a. 2

2

4

2 5

x x

x x

+

− b.

2

1

2 3

x

x x

+ − c.

2

2

2 8

4

x x

x

− −

2) a. 2 2

2 2

2 2 15

6 3 4

x x x x

x x x x

+ − + −

− − + − b.

2 2

2 2

9 2 3:

4 21 9 14

x x x

x x x x

− + −

+ − + +

c. 2 2

2 2

4 3 6 27:

4 5 14 45

x x x x

x x x x

− + + −

+ − + + d.

2

2

3

45

8 20

xx

x x

−+

+ −

3) a. 2

1 3 5

4 4 4x x x+ −

− + − b.

2

3 1 3

1x x x+ +

+ c.

2

1

11 30 5

x x

x x x

−+

+ + +

d. 2

2

1 1

6 7 7 1

x x

x x x x

++ +

− − − + e.

3 5 2

4 1x x x+ −

− + f.

( ) ( )3 2

1

2 3 2 3

x x

x x

++

+ +

Page 6: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 6

Roots (Zeros) of Polynomials Questions

1) Solve the following equations:

a. 3 22 2 0x x x− + − = b. 3 22 5 6 0x x x+ − − =

c. 3 22 5 6 0x x x− − + = d. 4 3 27 6 0x x x x− − + + =

e. 4 3 211 9 18 0x x x x− − + + = f. 3 27 14 8 0x x x− + − =

g. 3 2 17 15 0x x x+ − + =

2) Solve the following equations:

a. 3 22 3 8 3 0x x x+ − + = b. 3 22 2 1 0x x x+ − − =

c. 3 24 5 7 2 0x x x+ − − =

3) Solve the following equations (knowledge about the derivative is needed):

a. 3 2 5 3 0x x x+ − + = b. 3 3 2 0x x− − =

c. 5 4 3 23 5 27 32 12 0x x x x x− − + − + = d. 3 23 3 1 0x x x− + − =

4) Answer the following Questions

a. Solve the equation 3 2 2 2 0x x x− − + = .

b. Find all the zeroes of the polynomial ( ) 3 2 2 2= − − +p x x x x .

5) Answer the following Questions

a. Solve the equation 3 24 5 2 0x x x− + − = .

b. Find all the zeroes of the polynomial ( ) 3 24 5 2= − + −f x x x x .

6) Solve the equations:

a. 3 2 2 0− − − =x x x b. 4 3 25 3 6 0x x x x− − + + =

7) For each polynomial, list all its zeros and give their multiplicities:

a. ( ) 4 3 23 3 6p x x x x x= + + − − b. ( ) 5 4 3 25 9 9 8 4= − + − + −q x x x x x x

Page 7: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 7

Answer Key

1) a. 2x = 2,z i= b. 1,2, 3x = − − c. 1, 2,3x = − d. 1, 2,3x = −

e. 1,2, 3x = − f. 1,2,4x = g. 1,3, 5x = −

2) a. 1

1,3,2

x = − b. 1

1,2

x = − − c. 1

, 24

x = − −

3) a. 1, 3x = − b. 1,2x = − c. 1,2, 3x = − d. 1x =

4) a. 1, 2x = b. 1, 2x =

5) a. 1,2x = b. 1,2x =

6) a. 2x = , 1 3 1 3

,2 2 2 2

z i i= − + − − b. 1,2, 3x = −

7) a. 1, 2x = − , 1 2z i= − b. 1,2x = , z i=

Page 8: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 8

Partial Fractions Questions

Determine the partial fractions decomposition of the following expressions:

1) 2

1

4x − 2)

2

5

5

x

x x

+ 3)

2 5 6

x

x x+ +

4) 2

8 1

2 3 2

x

x x

− − 5)

( )2

4

1

+

x

x 6)

2

6

8 16

x

x x

+ +

7) 2

3

1x x

x x

+ −

− 8)

4 2

10

13 36

x

x x− + 9)

( ) ( )2

8

2 2− +

x

x x

10) 3 2

5 x

x x

+ 11)

3 2

9 36

6 9

x

x x x

+

+ + 12)

( )( )2 2

1

2 1 4 4x x x x− + − +

13) ( )

3

4

1

+

x

x 14)

( )

2

3

6 4 1

1

− +

x x

x 15)

( )( )

2

2

2 2 1

1 2

+ +

+ +

x x

x x

16) ( )( )

2

2

2 1

1 3

+ −

+ −

x x

x x 17)

( )( )2 2

3

1 4+ +x x 18)

( )2

2

1

1x x +

Answer Key

1) 1/ 4 1/ 4

2 2−

− +x x 2)

1 2

5−

+x x 3)

3 2

3 2−

+ +x x

4) 2 3

2 1 2x x+

+ − 5)

( )2

1 5

1 1x x+

− − 6)

( )2

1 10

4 4− +

+ +x x

7) 1 1/ 2 1/ 2

1 1+ −

− +x x x 8)

1 1 1 1

3 3 2 2+ − −

+ − + −x x x x 9)

2

1 4 1

2 ( 2) 2+ −

− − +x x x

10) 2

6 5 6

1− + +

+x x x 11)

( )2

4 4 3

3 3− −

+ +x x x

12) ( ) ( )

2 2

2 1 2 1

1 21 2+ − +

− −− −x xx x 13)

( ) ( )2 3

1 5

1 1+

− −x x

14) ( ) ( )

2 3

6 8 3

1 1 1+ +

− − −x x x 15)

2

1

1 2

x

x x+

+ + 16)

2

1 2

1 3x x+

+ −

17) 2 2

1 1

1 4−

+ +x x 18)

( )22 2

1

1 1− −

+ +

x x

x x x

Page 9: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 9

End Behavior of Polynomial Functions

Questions

1) Fill in the following table:

( )p x lim ( )x

p x→

lim ( )x

p x→−

2 3x− 5 2x− 7

23 5 7x x− − 2 43 5 7x x− − 33 5 7x x− − 2 53 5 7x x x− −

2) Match each polynomial function with the correct graph:

a. 3 2

2, 0y ax bx cx a= + + −

b. 2

2, 0y ax bx a= + −

c. 3 2

2, 0y ax bx cx a= + + −

d. 3, 0y ax a= −

e. 2

2, 0y ax bx a= + −

f. 3, 0y ax a= −

Page 10: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 10

3) Match each polynomial function with the correct graph:

a. 2 3y x= − +

b. 2 4y x x= −

c. 22 5y x x= − −

d. 32 3 1y x x= − +

e. 4 21

34

y x x= − +

f. 3 21 4

3 3y x x= − + −

g. 4 32y x x= +

h. 5 31 9

25 5

y x x x= − +

4) Match each polynomial function with the correct graph: a. 5y x= −

b. 2 4y x x= −

c. 4 23 5y x x= − +

d. 5 4 32 2 5 2y x x x x= − + − + −

e. 21

3 52

y x x= − + −

f. 3 2 3 6y x x x= − + −

1. 2. 3.

4. 5. 6.

1.

2.

3. 4.

5. 6.

7. 8.

Page 11: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 11

5) Match each polynomial function with the correct graph:

a. 3y x x= −

b. 3y x x= − +

c. 4 1y x= − +

d. 4y x=

e. 3y x=

f. 4 2y x x= −

Answer Key

1)

( )p x lim ( )xp x

→ lim ( )

xp x

→−

2 3x − − 5 2x− − 7 7 7

23 5 7x x− − 2 43 5 7x x− − − − 33 5 7x x− − − 2 53 5 7x x x− − −

2) a. v b. iv c. i d. vi e. iii f. ii 3) a. 6 b. 8 c. 3 d. 1 e. 5 f. 4 4) a. 1 b. 3 c. 2 d. 6 e. 5 f. 4 5) a. 1 b. 4 c. 2 d. 6 e. 5 f. 3

1.

2.

3. 4.

5. 6.

Page 12: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 12

Graphing Rational Functions

Questions

1) Find any vertical asymptotes for the graph of:

a. 2

2( )

9

xf x

x=

− b.

2( )

1

xf x

x=

+

c. 2

4( )

2 3

xf x

x x

+=

+ − d.

3

5( )

27f x

x=

2) Find any vertical asymptotes for the graph of

2

2

2( )

4

x xf x

x

+=

−.

Hint: is the rational expression in its lowest terms?

3) Find any horizontal asymptotes for the graph of:

a.

2

2

4( )

2 3

xf x

x=

− b.

2

( )4

xf x

x=

+

c. 2

3( )

2

xf x

x x=

− − d.

2

2

2 3 4( )

5 6 7

x xf x

x x

− +=

+ −

4) Find any oblique asymptotes for the graph of:

a.

2

( )3

xf x

x=

− b.

3

( )2 3

xf x

x=

c.

2 3 4( )

2 3

x xf x

x

− −=

− d.

3 2

2( )

2

x xf x

x x

+=

+ −

5) Sketch a graph of 3

( )3

f xx

=−

.

6) Sketch a graph of 2

1( ) 1f x

x= − .

Page 13: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 13

7) Sketch a graph of 4

( )2

xf x

x

−=

+.

8) Sketch a graph of 2

3( )

9

xf x

x=

−.

9) Sketch a graph of

2

2( )

1

xf x

x=

−.

10) Sketch a graph of

3

2( )

18 2

xf x

x=

−.

11) Sketch a graph of

2

2( )

2

xf x

x x=

+ −.

12) Sketch a graph of

2

2

4( )

1

xf x

x

−=

+.

13) Analyze and sketch a graph of

2

2

9( )

2 9 9

xf x

x x

−=

− +.

Note: the numerator and denominator here have factors in common.

Answer Key

1) a. 3x = b. There are no vertical asymptotes. c. 1, 3x x= = − d. 3x =

2) 2x =

3) a. 2y = b. There are no horizontal asymptotes. c. 0y = (the x-axis) d. 25

y =

4) a. 3y x= + b. There are no oblique asymptotes. c. 312 4

y x= − d. y x=

Page 14: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 14

5)

6)

7)

Page 15: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 15

8)

9)

10)

Page 16: Polynomial and Rational Functions Algeb… · Polynomial and Rational Functions Polynomial Division Questions: 1) Use long division to perform the following division problems: a

College Algebra

For more information and all the solutions, please go to www.proprep.uk. For any questions please contact us at +44-161-850-4375 or [email protected].

© All rights in this workbook reserved to proprep™ 16

11)

12)

13)