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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
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− +
College Algebra
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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−
College Algebra
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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
++
+ +
College Algebra
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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
++
+ +
College Algebra
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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
College Algebra
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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=
College Algebra
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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
College Algebra
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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= −
College Algebra
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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.
College Algebra
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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.
College Algebra
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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= − .
College Algebra
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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=
College Algebra
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5)
6)
7)
College Algebra
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8)
9)
10)
College Algebra
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11)
12)
13)