Name: _______________________________________________ Date: _________________ Pre- Calculus
UNIT 1- Algebra Extensions
Day 1-2 - Factoring
Difference of Two Perfect Squares, Trinomials,
Grouping, and Sum and Difference of Two Cubes
1. 100  9 y 2
2. 9 z 2  25w2
3. 81a 2  b 6
4. x 2  x  20
5. c 2  8c  12
6. x 2  16 x  36
7. a 2  11a  30
8. 2 x 2  16 x  30
9. x 3  8
10. z 3  216
11. z 3  125w3
12. 50 x 2  8
13. b 4  16
14. 125  5x 2
15. 3x 2  13x  10
16. 4 x3  12 x 2  16 x
17. 3x 2  10 x  8
18. x 4  3 x 2  2
19. x 2 n  x n  2
20. 12  5 x  2 x 2
21. x( x  10)  4( x  10)
22. x 2 (a  2)  a  2
23. x n 1  3 x n
24. 3x3n  9 x 2 n
2
1
2
1
25. x 3  3x 3 10
26. 3x 3  8x 3  5
27. 125  27x 3
28. 2 x 2 n  x n  3
29. ( x 2  1)2  ( x 2  1)  6
30. 4 x 2  x3  3x 4
Day 3– Rational Expressions
Undefined Expressions, Simplifying Rational Expressions, Multiplying and Dividing Rational
Expressions
1. Find all values of x that make the fraction
2. Find all values of x that make the fraction
2x  7
undefined:
3 x  3 x  18
2x
undefined:
x 3
2
3. Find all values of x that make the fraction
x 3  x 2 12x
x3  4x2
4. Simplify:
16  x 2
undefined:
x3  2 x 2  8 x

6x 2  5x  4
6. Reduce
to lowest terms
8  2x  3x 2
x2y2  9
5. Simplify:
3  xy
2x  4
7. The length of a rectangle is 2
and its width is
x 9
x 
3
. Express the area of the rectangle as a single
2
fraction in simplest form.

9. Simplify:
x 2  25 x 2  8x  12 x 2  7x  10


2x  12
4 x  20
8x
9x 2
2x 2  5x  2

8. Simplify:
3x 2  6x
6x  3


10. Multiply:
x3  8
x3  2 x 2

x3  2 x 2  4 x x 2  4
2x  6
4 x 2 100
20  4 x
 2
 2
11. Simplify: 2
x  6x  5 x  x  6 2x  9x  10

12. Divide:
27 x3  8
9 x 2  12 x  4

9 x3  6 x 2  4 x
9 x2  4
x2n  xn  2 x2n  xn
 2n
x2n  xn
x 4
14. Divide:
8 x3  12 x 2 y
16 x 2 y 2

4 x2  9 y 2
4 x 2  12 xy  9 y 2
13. Multiply:

15. Find all values of x and y that make the fraction
6 x3  15 x 2  75 x
150 x  30 x 2  12 x3
undefined:
16. Divide:
x4n  1
x2n  1

x 2 n  x n  2 x 2 n  3x n  2
Day 4 – Rational Expressions
LCDs, Adding and Subtracting Rational Expressions, Simplifying Complex Fractions
1. What is the sum of

3. Simplify:
3
x
and
?
x3
3 x
2
3

3m  9 m  3
3
y2
a 8

7. Simplify: 2 a
1 1

4 a
2a  2b
2 2

a b

4. Simplify:
1
1
 2
2x  2 x 1
6. Simplify:
6
y5
 2
y 5
y  25
4
x
8. Simplify:
16
1 2
x
1
10. Simplify:
8
5
4


9 x  4 3x  2 3x  2
x
2x  5
4
x3
x2
11. Simplify:
6x
2
3x


3x  7 x  2 3x  1 x  2
12. Simplify:
13. Simplify:
3x 2  2 9 x  x 2
 2
x2  4
x 4
14. Simplify:
2
5 6

y y2
3
1
y
1
15. Simplify:

2x  4
and its width is
x2  9
3
. Express the perimeter of the rectangle as a
x3
single fraction in simplest form.


5. Simplify: ( y  5) 
9. Simplify:
2. The length of a rectangle is
2
x2
5 6
1  2
x x
6
x 1
16. Simplify:
12
x 3
x 1
x6
Day 5-6 – Radicals & Conjugates
Simplifying, Adding and Subtracting, Multiplying and Dividing
and Rationalizing Denominators using the Conjugate
1. Simplify:
3
3. Simplify:
18x 4 y 3 z 5
2. Simplify: 3 x 60 x 3 y 7 m 2
75x 5 y 9
4. Subtract and Simplify: x 75 xy  27 x 3 y
5. Add and Simplify: 2a 27ab5  3b 3a3b
6. Simplify:
7. Add and Simplify: 3 3 3a4  3a 3 81a
8. Multiply:
9. Divide:
84 65ab 4
7 5ab
11. Multiply:
2
10. Divide:

3x  y 2 3x  y

13. Express with a rational denominator:
15. Divide and Simplify:
17. Simplify: 27
2
12. Multiply:
5
2 7
40 x3 y 2
4 x 7 y 5  9 x 2 x 3 y 5  5 xy x 5 y 3
4
12ab3
4
4 a 5b 2
42a3b5
14a 2b
2x

8 x  32
14. Express with a rational denominator:
16. Divide and Simplify:
80 x 2 y 3
3
18. Simplify:
19. Simplify using positive exponents only:
15 x 3 y 6
5a 6 y 4
21. Simplify using positive exponents only:

3 x
3 x
3
4
8x 3
(2 x 4 )3
(3x 2 )4
20. Simplify w/ positive exponents
4

1
2
22. Simplify: x 3 x 3  x 3
9a 2b6

1
 4a 4 b 6  2

1 
 25a b 
23. Simplify using positive exponents only:
24. Simplify:
3

2
 49c 3 
  1 5 
6
4
a b 
25. Simplify with a rational denominator:
5
 x  x   12x 
2
3
 16
3 0
26. Simplify with a rational denominator:
3
y 1 1
2
x42
3
27. Simplify and convert to radical form:
b4
b
 32
29. Simplify and write with rational exponents:
3
27x 9
x4
28. Simplify and convert to radical form:
x y 
n
8
30. Simplify and write with rational exponents:
49a 8
b5
n
8
2
Day 7-9 – Partial Fractions (Decomposition of Fractions)
Example: Decompose the following expression as the sum of two fractions:
Step 1: Factor the Denominator:
2x  3
x  2x  8
2
2x  3
( x  4)( x  2)


A
B
A
B




( x  4) ( x  2)
 First Factor Second Factor 
2x  3
A
B
Step 3: Create and equation using steps 1 and 2:


( x  4)( x  2) ( x  4) ( x  2)
Step 2: Create a sum of fractions in the following form:
Step 4: Eliminate the denominators by multiplying by the LCD: 2 x  3  A( x  2)  B( x  4)
Step 5: Use all values of x that would make the original fraction undefined in order to solve for A and B:
x  4
2(4)  3  A(6)  B(0)

Let x  4 
11  6 A
 A  11/ 6
 x  2
2(2)  3  A(0)  B(6)

Let x  2 
1  6 B
 B  1/ 6
Step 6: Having found the values of A and B, substitute those values back into the expression from Step 2:
11/ 6
1/ 6

( x  4) ( x  2)
For each of the following, decompose the given expression into the sum of two fractions:
1.
5x  1
x  x2
2.
7x  4
x  x6
3.
x 1
x  x6
4.
x5
x2 1
5.
5x  3
3x  4 x  1
6.
21x  5
2x2  x 1
7.
2x 1
x  3 x  18
8.
x 3
x  4 x  12
2
2
2
2
2
2
UNIT REVIEW:
1. Factor: 2 x 2  32 x  72
2. Factor: x 2 ( x  3)  4( x  3)
11. Divide:
6  2x
4x

2
x  9 15  5x

ax  x  ay  y
a 2 1

12. Divide:
y2  x2
2x  2y
3. Factor: 27 x3  125 y 9

13. Write as a single fraction
2
1
 2
a  4 a  2a

14. Simplify as one fraction:
5x
3x  2
4
2x  3
2x  3
2
4. Factor: 3 x 2  46 x  15
5. Factor: ax  bx  ay  by
6.Factor: 10y 2 19y  6

7. Factor: 2 x 2 n  18 x 2 n
8. Find all values of x that make the fraction
2x  3
undefined.
x  6 x  16
15. Subtract:
x
16. Simplify: x  3
3
x
2
3
x4
17.
x
3
x4
x
9. Find all values of x that make the fraction
x 2  49
undefined.
5 x 2  34 x  7
10. Multiply:

a 2  5a  4 2a  4
 2
3a  6
a 16
2x
x2

x 9 x3
2
x

y
18.
y

x
y
x
x
y
19. Simplify: 4 363x 5 y 7
20. Simplify:
3
 64x
29. Simplify w/ positive exponents:   1 5
 x 3 y6

5
3




1
2
375x 5 y 6
30. Simplify:
 m   m    2m 
5
3
 13
3 0
21. Simplify: x 600  2 24 x2  4 x 96
31. Simplify with rational exponents:
22. Simplify:
3
75x8
x2
162a4b3  3  ab 18a 2b 1
12 xy 3
23. Multiply and Simplify:

3xy  3

9x 1
 x  3 x  1
x 
25. Divide and Simplify:
18c 3
9c
33.
10 x  7
2 x  13 x  15
27 x3  3 36 x5
34.
2x 1
x  3x  4
3
26. Divide and Simplify:
3
3 y 2x  3 y
4

32.
24. Multiply and Simplify:
4

32-34: Decompose the given expression into the
sum of two fractions:
3x3
27. Simplify w/ a rational denominator:
7
7y
28. Simplify w/ a rational denominator:
5x
5x  2
2
2
IMPORTANT FORMULAS AND THINGS TO RECALL FROM ALGEBRA