Unit 1 Homework Problems MAT150 College Algebra
© Andrew McKintosh 2010
All problems should be done WITHOUT a calculator unless otherwise indicated!!!!
Day 2 Homework: Topics: Sets of Real Numbers, Math symbols (equality and inequality), Interval notation and
graphing inequalities, Absolute value, Terms and coefficients.
Supplemental Reading: 1. http://www.jamesbrennan.org/algebra/numbers/real_number_system.htm
2. http://regentsprep.org/REgents/math/ALGEBRA/AP1/IntervalNot.htm
1. List each of the following numbers next to the set (or sets) of real numbers it belongs in.
6
3
5,  , , 7,  25, , 0, 25, .134, .12, 4
3
5
6
Natural Numbers:
, 4
3
6
Whole Numbers: , 0, 4
3
6
Integers: 5, ,  25, 0, 4
3
6
3
Rational Numbers: 5, ,  25, , 0, .134, .12, 4
3
5
Irrational Numbers:  , 7
6
3
Real Numbers: 5,  , , 7,  25, , 0, .134, .12, 4
3
5
2. On the line next to each problem, list all “symbols” that could be placed inside the box to produce a true
statement. Choose from the following symbols….. , , , , , 
3
5
, , 
3
1
51

17 , , 
3
4
3. Complete the following table (just like we did in class!)
Inequality
Interval
, , 
5

1
 , 
2
Graph
 , 2
x2
2
 1, 4
1  x  4
-1
4
3,  
x3
3
4. Simplify the following absolute value statement. Recall the definition of absolute value
5
x2  1 =
4 
5
x2  1
0 
4
6  3x
if x  2
0
3x  6
5. List the terms for the following expression (keep the sign with the term)
4 x3  7 x 2  5x  3
4 x3 , 7 x2 ,  5x , 3
For each term above, list the coefficient
4 , 7,  5 , 3
 w if w  0
w 
  w if w  0
Day 3 Homework: Topics: Properties (rules) of Algebra; , , ,  fractions
© Andrew McKintosh 2010
Supplemental Reading: 1. http://www.analyzemath.com/algebra/rules_algebra.html
2. http://www.math.com/school/subject1/lessons/S1U4L3GL.html
1. Identify the rule(s) of Algebra that are being illustrated by the following statements.
A. 3  x  y    x  y  3 Commutative Property of multiplication
B. 3 x  y   3x  3 y
Distributive Property
C. 3  x  y   3 y  x  Commutative Property of Addition
D.
1
1
 5  6    5  6
5
5 
Associative Property of Multiplication
1 
E.   5  6  1 6
5 
Multiplicative Inverse Property
F. 1 6  6
Multiplicative Identity
G.
 x  y  0   x  y
Additive Identity
2. Simplify the following…..
A.
0
 0
5
B.
5
 undefined
0
C. 5  0 
0
D. 5  0 
5
3. Simplify the following…..
A.
17 32 17  32 15



 3
5
5
5
5
C.
8 6
8 25 4 5 20

 
  
15 25 15 6 3 3 9
B.
5 7 11 30 56 99 30  56  99 86  99
13
  





12 9 8 72 72 72
72
72
72
Day 4 Homework: Topics Rules of exponents and simplifying.
© Andrew McKintosh 2010
Fractional exponents, radicals and simplifying.
Supplemental Reading:
1. http://www.purplemath.com/modules/exponent.htm (skip scientific and engineering notation)
2. http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_radical_simplify.xml
3. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut5_ratexp.htm
4. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut2_exp.htm
5. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut4_radical.htm (up through
example 11)
1. Evaluate
3
A) 23  32
B) 2 4
1 1

2 3 32
1 1

8 9
9
8

72 72
1
72
 3 5
D)     
 5 3
33 52
 3 2
5 3
33 2
 3 2
5
3

5
C) (2)4
16
16
2. Simplify
2 x 2   3 y 2 
 3 yz 
C)
2y z 
2 y z 
 3 yz 
2  y  z 
 3  y  z 
3 3
3
A)
 3x   9 x 
2 3
 3
3
4 1
B)
1
1 2
3 2 4
 9 y 1
2 x 2  33  y 2 
 x  9  x 
2 3
 2x 
4 1
3
3 2 4
22  x 1   9 y 1
2
1
27  x 6   x 4
9
27 x 6 4

9
3 x 2
3 3
2 x 2  27 y 6
22 x 2 9 y 1
4
2  2  3 y  y
x2  x2
24 y 7
 4
x
2
6
1
3 4
2 4
3
3 3
3
16 y 12 z 8
27 y 3 z 9
16

3
27 y  y12 z 9 8
16

27 y15 z
3. Simplify. Assume all variables represent non-negative real numbers.
 8 
A)  
 27 

1
3
B) 32
3


3
2
27
8
3
5
1
1
 27  3
 
 8 

3 xy 2 6 x
32
1
323
1
32  32  32
1
222
1
8
5
54x3 y 4
9  6 x3 y 4
3
5
5
C)
D)
3
3
54x3 y 4
27  2 x 3 y 4
3 xy 3 2 y
2
Day 5 Homework: Topics Combining Radicals, Rationalizing the denominator © Andrew McKintosh 2010
Supplemental Reading:
1. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut4_radical.htm (after example 11)
1. Simplify. Assume all variables represent non-negative real numbers.
A) 2 50  14 8
B) 4 27  75
C) 3 3 16  5 3 54
2 25  2  14 4  2
4 9  3  25  3
3 3 8  2  5 3 27  2
2  5 2  14  2 2
43 3  5 3
3 2 3 2  5  33 2
10 2  28 2
12 3  5 3
6 3 2  15 3 2
18 2
7 3
213 2
D) 7 x 80 x  2 125x3
E) 9 3 24 x 6 y 3  5 xy 3 192 x3
7 x 16  5 x  2 25  5  x 3
9 3 8  3x 6 y 3  5 xy 3 64  3x3
7 x  4 5x  2  5  x 5x
9  2 x 2 y 3 3  5 xy  4 x 3 3
28 x 5 x  10 x 5 x
18 x 2 y 3 3  20 x 2 y 3 3
18 x 5 x
2 x 2 y 3 3
2. Rationalize the denominator then simplify your answer if possible.
6
A)
10
6
10
C)
B)

10
10

6 10
100
6 10 3 10

10
5
4
3
2
2
3
4
3
4

43 4
3
8

43 4
 23 4
2
6
3 5
6

3 5
3 5 3 5

18  6 5
9  25



18  6 5 18  6 5 2 9  3 5
93 5



95
4
4
2
10
D)
8 3
10
8 3
E)

4
3

8 3
8 3

10 8  10 3
64  9



10 4  2  10 3 10  2 2  10 3 20 2  10 3 5 4 2  2 3



 4 2 2 3
83
5
5
5
2 3
3 3
2 3 3 3 6 2 3 3 3  9 65 3 3 95 3




93
6
3 3 3 3
9 9
Day 6 Homework: Topics Polynomials , ,  : Find shaded area.
© Andrew McKintosh 2010
Supplemental Reading:
1. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htm
1. For each polynomial list the following…
A) Name (Monomial, Binomial, Trinomial or Polynomial)
B) Write the polynomial in descending order.
C) List the degree of each term
D) List the degree of the polynomial
E) Write down the lead coefficient
I. 4x  3x3  x2
II. 4  2x  x3  2x2
A) Trinomial
A) Polynomial
B) 3x3  x2  4x
B) x3  2x2  2x  4
C) 3, 2, 1
C) 3, 2, 1, 0
D) 3
D) 3
E) 3
E) 1
III.
A)
B)
C)
D)
E)
4  x2
Binomial
 x2  4
2, 0
2
1
2. Add, subtract or multiply as indicted.
A) 2  x 2  5 x  6    x3  2 x  5   4  x3  4 x 2  2 
IV.
A)
B)
C)
D)
E)
5x
Monomial
5x
1
1
5
B) 3x3  2 x 2  5 x  3
6x5  15x4  9x3
2 x 2  10 x  12  x3  2 x  5  4 x 3  16 x 2  8
3x3  14 x 2  12 x  25
C)
3x  5 2x  7 
6 x 2  21x  10 x  35
 3x  2 
 3x  2  3x  2 
6 x 2  11x  35
9x2  6x  6x  4
D)
2
9 x 2  12 x  4
3. Find the area of the shaded region. Write your answers as a polynomial with terms in descending order.
2x + 1
A)
4x + 2
B)
2x – 1
2x - 3
5x – 2
x+1
x+1
3x – 2
Shaded area  big area  small area
  5 x  2  4 x  2    2 x  3 x  1
 20 x 2  10 x  8 x  4   2 x 2  2 x  3x  3
 20 x 2  10 x  8 x  4  2 x 2  2 x  3x  3
 18 x 2  3x  1
Shaded area  big area  small area
1
1
 bh  bh
2
2
1
1
  2 x  1 3 x  2    x  1  3 x  2    2 x  1 
2
2
1
1
  6 x 2  4 x  3 x  2    x  1 3 x  2  2 x  1
2
2
1
1
  6 x 2  x  2    x  1 x  1
2
2
1
1 2
2
 3x  x  1   x  1
2
2
1
1 2 1
2
 3x  x  1  x 
2
2
2
5 2 1
1
 x  x
2
2
2
Day 7 Homework: Topics Factoring polynomials
© Andrew McKintosh 2010
(GCF, Binomials (Diff of squares), Trinomials)
Supplemental Reading:
1. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm (Note: Skip the
section on Grouping (ex 3 & 4) and the section on Sum & Difference of Cubes (ex 14 & 15))
Factor completely:
1. x2  6 x  16
 x  8 x  2
2. 9x2 16
 3x    4 
 3x  4  3x  4 
2
2
3. 6 y 2  54
4. x2 10x  21
6  y2  9
 x  7 x  3
6
 y 
2
  3

2
6  y  3  y  3 
5. 6x2  12 x  6
6  x 2  2 x  1
6. 5a3  25a2  15a
5 a  a 2  5a  3 
6  x  1 x  1

5  x 2   1
or
6  x  1
7. 5x 4  5
5  x 4  1
2
2
8. x2  7 x  18
 x  9 x  2

5  x 2  1 x 2  1
2

5  x 2  1  x   1
2
2

5  x  1  x  1 x  1
2
9. a2  12a  27
 a  9 a  3
10. 49x2 1
11. x4  81
 7 x   1
 7 x  1 7 x  1
2
2
12. 4 x 2  25 y 2
 x   9
 x  9  x  9 
 x  9    x    3 
 x  9   x  3 x  3
2 2
2
2
2
2
2
 2x   5 y 
 2 x  5 y  2 x  5 y 
2
2
2
2
13. 4 y 2  7 y  2
 4 y 1 y  2
17. 12x2  4x  5
 6x  5 2x 1
14. 4 y 2  17 y  15
 4 y  3 y  5
15. 4 y 2  4 y  15
 2 y  3 2 y  5
16. 6x2  7 x  2
3x  2 2x 1
Day 8 Homework: Topics Factoring polynomials
© Andrew McKintosh 2010
(Grouping (4 terms), Binomials- Sum and Diff of cubes)
Supplemental Reading:
1. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm (Read the
sections that I told you to skip yesterday!)
Factor completely:
1. x3  2x2  5x  10
y 3  y 2  3  4  y 2  3
x2  x  2  5  x  2
 x  2   x 2  5
y
4. x3  3x2  4x  12
 x  3  x 2  8 
 3 y 3  4 
6. x 2  y 2  6 y  9
x2   y 2  6 y  9
 x  2  x  2   y 2
2
 x  2  y2
 x  2  y  x  2  y 
 x  3  x 2  4 
 x  3   x    2  
 x  3 x  2  x  2 
2
x 2   y  3 y  3
x 2   y  3
9. 64a3  27b3
8. 250 y 3  16
 5 x   1
2
2
 5 x  1   5 x    5 x 1  1 
3
 5 x  1  25 x
2
 5 x  1
2 125 y 3  8 

2 5 y    2
3

 4a    3b 
2
2
 4a  3b    4a    4a  3b    3b  
3
3

2  5 y  2   5 y    5 y  2    2 
2
2
2  5 y  2   25 y 2  10 y  4 

3
 4a  3b  16a 2  12ab  9b 2 
11. x6 1
10. 2 x 6  54 y 6
2  x 6  27 y 6 

2
 x  y  3  x   y  3 
 x  y  3 x  y  3
7. 125x3 1
3
2
x 2  x  3  8  x  3
5. x 2  4 x  4  y 2
x 2  x  3  4  x  3
2
3. x3  3x2  8x  24
2. y 5  3 y 3  4 y 2  12
2  x2   3 y 2 
3

 x   1
 x  1 x  1
3 2
3

2
3
2  x 2  3 y 2   x 2    x 2  3 y 2    3 y 2 
2
2  x 2  3 y 2  x 4  3x 2 y 2  9 y 4 
2

3
 x   1   x   1 
3
3
3
3
 x  1  x 2  x  1  x  1  x 2  x  1
Find the area of the shaded region. Write your answer in factored form!
1.
R
2.
R
R+2
Shaded area  Big area  Small area
Shaded area  Big area  Small area
   R  2   R2
  2 R  2 R    R 2
2
   R  2   R 2 


   R  2  R  R  2  R 
 4R2   R2
2
   2 R  2  2 
   2  R  1 2 
 4  R  1
 R2  4   
Day 9 Homework: Topics Finding the domain of a rational expression,
© Andrew McKintosh 2010
Simplifying rational expressions
Supplemental Reading:
1. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htm
Factor completely:
1. x2 11x  42
 x 14 x  3
2. 6x2  19 x  7
3x 1 2x  7
3. 2 x4  128
2  x 4  64 

2  x 2   8
2
2

2  x 2  8  x 2  8 
4. 2 x6  128
2  x 6  64 

5. 3x4  6x3  48x2  96x
2  x3   8
2
2
3 x  x 3  2 x 2  16 x  32 

3 x  x 2  x  2   16  x  2  

3 x  x  2   x 2  16 
2  x3  8  x3  8 

2  x    2
3
3
  x    2 
3


3x  x  2   x    4 
3
2
2

3 x  x  2  x  4  x  4 
2  x  2  x2  2x  4  x  2  x2  2x  4
Find the Domain of the following expressions. Write your answer using interval notation.
2x 1
x4
x4 0
1.
2.
3  6x
3  6x  0
6 x   3
D :  , 4    4,  
3
x
6
1
x
2
1

D :  , 
2

Write the rational expression in simplest form.
x  4
27 x 4
63x7
9  3x 4
9  7 x7
3
7 x74
3
7 x3
1.
4.
2.
D :  , 
3.
2  x  2 
2 x
x2  5x  6
x2  9
 x  2  x  3
 x  3 x  3
x2
x3
4 x2  8x
2 x  4
4x  x  2
3. x2  x  6
x3
5.
 x  10  x  1
x2
x 1
 x  6
x  6 x  6
x2
x 2  8x  20
x 2  11x  10
 x  10  x  2 
x  10
x 2  36
x6
 x  6  x  6 
6.
x3  5 x 2  10 x  50
x 2  3x  10
2
x  x  5   10  x  5 
 x  5 x  2 
 x  5  x 2  10 
 x  5 x  2 
x 2  10
x2
x5
Homework Day 10: Topics Multiplying and Dividing Rational Expressions
© Andrew McKintosh 2010
Supplemental Reading: 1. http://www.purplemath.com/modules/rtnlmult.htm
2. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm
Multiply or Divide as indicated.
x2  x  6 x2  4
1. 2

x  6x  9 x  3
2.
x2  x  6 x  3

x2  6x  9 x2  4
 x  6  x  6 
x
 x  3 x  2 
 x  3

 x  3 x  3  x  2  x  2 
 x  3
 x  3 x  2 
3.
x 2  36 x 2  x
 3
x
x  6x2

 x  6  x  1
x2
x  x  1
x2  x  6
x6
x  2
x  1 x2  9

x  3 x2  x  2
4.
y3  1 y 2  y  1

y2 1 y2  2 y  1
 x  1  x  3 x  3

 x  3  x  2  x  1
y3  1 y 2  2 y  1

y2 1 y2  y 1
x3
x2
 y  1  y 2  y  1  y  1 y  1

 y  1 y  1
 y 2  y  1
x  1,3
y  1 y  1,1
5.
x 2  x  6 x 2  3x  10

x 2  7 x  12 x 2  2 x  8
6.
x 2  x  12 x 2  x  6

x 2  6 x  8 x 2  2 x  24
x2  x  6 x2  2x  8

x 2  7 x  12 x 2  3 x  10
 x  4  x  3  x  3 x  3

 x  4  x  2   x  6  x  4 
 x  2  x  3  x  2  x  4 

 x  4  x  3  x  2  x  5 
 x  3 x  3
 x  6  x  4 
x2
x 5
OR
x  2,3, 4
 x  3
 x  6  x  4 
x  2, 4
2
x  2, 4
Homework Day 11: Topics Adding and Subtracting Rational Expressions
© Andrew McKintosh 2010
Supplemental Reading: 1. http://www.purplemath.com/modules/rtnladd.htm
2. http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut10_addrat.htm
Add or subtract the rational expressions as indicated. Always simplify your answer if possible.
5
1
12
8x
5
3
1.
2.
3.



2x 2x
2x  3 2x  3
4 a 8a
8
x

12
2
5
3
5 1

2x  3
2  4a 8a
2x
10
3
4  2 x  3
4

8a 8 a
2x
 2 x  3
7
2
3
4
x


8
a
x
2
4.
x2  4x  4 x  4

2 x 2  3x  1 2 x  2
x2  4x  4
x4

 2 x  1 x  1 2  x  1
2   x2  4x  4
x
5

x 2  11x  30 x 2  9 x  20
x
5

 x  5 x  6   x  5  x  4 
5.
x  x  4
 2 x  1 x  4 
2   2 x  1 x  1 2  2 x  1 x  1

 x  5 x  6  x  4   x  5  x  6  x  4 
x 2  4 x  5 x  30
 x  5 x  6  x  4 
2 x2  8x  8  2 x2  8x  x  4
2  2 x  1 x  1
x 2  x  30
 x  5 x  6  x  4 
4x2  x  4
2  2 x  1 x  1
6.
5  x  6

 x  6  x  5
 x  5 x  6  x  4 
9x  2
7

3x 2  2 x  8 3x 2  x  4
9x  2
7

 3x  4  x  2   3x  4  x  1

x6
 x  6  x  4 
3z
2z

z 2  7 z  10 z 2  8 z  15
3z
2z

 z  5 z  2   z  5  z  3
7.
3 z  z  3
2z  z  2
7  x  2
 9 x  2  x  1

 3x  4  x  2  x  1  3x  4  x  2  x  1
 z  5 z  2  z  3  z  5  z  2  z  3
9 x  9 x  2 x  2  7 x  14
 3x  4  x  2  x  1
3z 2  9 z  2 z 2  4 z
 z  5 z  2  z  3
9 x 2  16
 3x  4  x  2  x  1
z 2  5z
 z  5 z  2  z  3
2
 3x  4  3x  4 
 3x  4  x  2  x  1

3x  4
 x  2  x  1
x
4
3
z  z  5
 z  5 z  2  z  3
z
z

2

 z  3
x
3

x 1 1 x
x
3

x  1   1  x 
8.
x
3

x  1 1  x
x
3

x 1 x 1
x3
x 1

z5
x  5
Homework Day 12: Topics: Plotting points, Distance formula,
© Andrew McKintosh 2010
Midpoint Formula, Pythagorean Theorem
Supplemental Reading:
1. http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut20_rect.htm
2. http://www.purplemath.com/modules/pythagthm.htm
3. http://www.purplemath.com/modules/distform.htm
1. Plot the following points on the given graph.
A
C
A 2,5 B 3, 2 C 1, 4 D  3, 2
D
B
2. Find the distance from point A to point B
d 
 x2  x1    y2  y1 
2

 3  2    2  5

 5   7 
2
2
 25  49
2
2
2
 74
3. Find the coordinates of the midpoint of the line segment joining points A and B
 x  x y  y2
midpoint   1 2 , 1
2
 2
  2  3 5  2   1 3 
  2 , 2  2,2
 

 
4. Find the distance from point C to point D
d 
 x2  x1    y2  y1 
2

 3  1   2  4 

 4    6 
2
2
2
2
2
 16  36  52  4 13  2 13
5. Find the coordinates of the midpoint of the line segment joining points C and D
 x  x y  y2   1  3 4  2   2 2 
midpoint   1 2 , 1

,
   ,    1,1
2   2
2   2 2
 2
6. Is a triangle with side lengths 8, 15 and 17 a right triangle?
82  152  17 2 ?
64  225  289 ?
289  289 YES !!!!
7. A right triangle has its two shortest sides with lengths 7 and 24, what is the length of the longest side?
7 2  242  c 2  49  576  c 2  625  c 2
c   625  25
c  25