72
A2T Packet #1: Rational Expressions/Equations Name:______________________________ Teacher:____________________________ Pd: _______
A2T
Packet #1: Rational Expressions/Equations
Name:______________________________
Teacher:____________________________
Pd: _______
Table of Contents o Day 1: SWBAT: Review Operations with Polynomials
Pgs: 1-3
HW: Pages 2-3 in Packet
o Day 2: SWBAT: Factor using the Greatest Common Factor (G.C.F.) Pgs: 4-8
HW: Textbook page 26-27 (Page 9 in Packet) #3 – 8, 9 - 14, 27, 29, 31, 32, 33, 37, 38
o Day 3: SWBAT: Factor quadratic trinomials of the form ax2 + bx + c. Pgs: 10-14
HW: Homework: Textbook page 26-27 (Page 15 in Packet) #15 – 25 (odd), 28 - 40 (even), 42 – 45
o Day 4: SWBAT: Review of Factoring
Pgs: 16-17
HW: Pages 18-19
o Day 5: SWBAT: Simplify Rational Expressions
Pgs: 20-25
HW: Homework: Textbook page 47 – 48 (Page 26 in Packet) # 6, 7, 8, 10, 14, 15 – 27 odd
o Day 6: SWBAT: Multiply and Divide Rational Expressions
Pgs: 27-32
HW: Textbook pages 52 – 53 (Page 33 in Packet) # 3 – 29 odd
o Day 7: SWBAT: Adding and Subtracting Rational Expressions with Like Denominators
Pgs: 34-38
HW: Page 39 in Packet
o Day 8: SWBAT: Adding and Subtracting Rational Expressions with Unlike Denominators
Pgs: 40-44
HW: Textbook Pages 56 – 57 (Pages 45-46 in Packet) #’s 3 – 23 odd and Page 45
o Day 9-10: SWBAT: Simplify Complex Fractions
Day 9: Pgs: 47-52
Day 9: HW: Textbook Page 64 (Page 53 in Packet) #’s 7 – 23 odd;
Day 10: Pages 54-55
o Day 11-12: SWBAT: Solve Rational Equations
Day 11: Pgs: 56-60
HW: Pages 61-62 in Packet #1 – 47 every other odd
Day 12: Pages 63-65
o Review – Pages 66 - 67
o Practice Test - Pages 68 - 70
TEST 1: ________________________
1
Day 1: Operations with Polynomials
A monomial is a constant, a variable, or the product of constants and variables.
Ex. 3, a, ab, -2a2
3a4; 3 is the coefficient , a is the base and 4 is the exponent.
A polynomial is the sum of monomials. Each monomial is a term of the polynomial.
Ex. 3a2 + 7a - 2
When a
**When adding and subtracting polynomials, add or subtract the coefficients of like terms.
Ex. 3a2 + 5a
2 = 8a
2
( 3 5 + 9) + ( 3 3
) = -2 3 4 + 9
***Remember , when you subtract you must change the signs!
Ex. Subtract (3b4 + b + 3) from (b
4 5b +3).
(b4 5b +3) - (3b
4 + b + 3)= (b
4 5b +3)+ (-3b4 b 3)= -2b
4 6b
**When multiplying monomials, multiply the coefficients and add the exponents of like bases.
Ex. (3a2b)(2abc) = 6a
3b
2c
Ex. ab(a2 + 2ab + b
2) = a
3b + 2a
2b
2 + ab
3
Ex. (-2x3y)
2 = (-2x
3y)(-2x
3y) = 4x
6y
2
Ex. FOIL!!!! (3x-2)(2x+5) = 6x2 +15x – 4x – 10 = 6x
2 + 11x -10
Multiplying Monomials and Polynomials
Adding and Subtracting polynomials
2
HW #1: Operations with Polynomials: Write your answers in simplest form.
1) (3y – 5) + (2y - 8) =
2) (x2 + 3x – 2) + (4x
2 – 2x + 3) =
3) (7b2 – 2b + 3) – (3b
2 + 8b + 3) =
4) (4x2 – 3x – 5) – (3x
2 – 10x + 3) =
5) 2a5b
2(7a
3b
2) =
6) (6xy2)2 =
7) 2x2y(y – 2y
2) =
8) (x + 3)(2x – 1) =
9) (a + 3)2 =
10) (2x + 3)(x2 + x – 5) =
11) a3(a
2 + 3) – (a
5 + 3a
3) =
12) The length of a rectangle is 4 more than twice the
width, x. Express the area of the rectangle in terms of
x.
3
Solving Equations and Inequalities
1) 5x + 4 = 39
2) 7a + 3 > 17
3) 7x + 5 = 4x + 23
4) (b – 1) – (3b – 4) = b
5) 2(b – 3) + 3(b + 4) = b + 14
6) -3 – 2x ≥ 12 + x
7) 4x(x + 2) – x(3 + 4x) = 2x + 18
8) 5y – 1 ≥ 2y + 5
4
Day 2: Factoring by GCF SWBAT: Factor polynomials by using the GCF.
Warm – Up
There are 4 types of Factoring Techniques for the unit.
o
o o o
Greatest Common Factor (GCF)
Step 1: Find largest number that divides into ALL terms.
Step 2: Find variables that appear in ALL terms and pull out the smallest exponent for that variable. Step 3: Write terms as products using the GCF as a factor.
Step 4: Use the Distributive Property to factor out the GCF.
Step 5: Multiply to check your answer. The product is the original polynomial.
Example 1: Factor using GCF:
5
Practice: Factor each polynomial using the GCF and check your answer.
a. 7n3 + 14n + 21n2
b. a2
b3 + ab
2c
c.
Example 2: Factoring a common binomial factor Using the GCF
2) 4x(x + 1) + 7(x + 1) 3) y(y – 2) - (y – 2)
Practice: Factor each polynomial and check your answer.
d) e) f)
Factor by Grouping Use when more than three terms
o Group the terms and factor each group
o Factor out the common term (a + b)
o Answer will be written in the form of: (a + b)(c + d)
6
Example 3:
Practice: G H
7
Difference of Two Squares (DOTS)
o Binomial
o Both terms are perfect squares
o Even exponents on variables Divide exponent by 2
o Perfect squares for coefficients Square root coefficient
o Pattern: –
= (x + y)(x – y)
Example 4: Factor the binomial below.
– Practice: Factor each of the binomials below. I. 4x
2 – 25y
2
J. 16 – 9n8
K. ** 36a
4 – b
4
L. ** 27x5 – 75x3
8
Challenge Problem:
Summary:
Exit Ticket:
9
Day 2: Homework:
Homework: Textbook page 26-27 #3 – 8, 9 - 14, 27, 29, 31, 32, 33, 37, 38
Homework Answers
10
Day 3: SWBAT: Factor quadratic trinomials of the form ax2 + bx + c. Warm – Up
Trinomials
o Has three terms ax2 + bx + c
o Must find numbers to multiply to equal ac and add to equal b
o Once you find these numbers, you can use grouping/rainbow method to rewrite the problem and finish it
Example 1: Factoring Polynomials of the form ax
2 + bx + c
1) trinomials where a = 1
Example: x2 + x - 6
Split the x2: (x )(x )
Look for two numbers that multiply to -6 (c) and add to +1 (b): +3, -2
Final answer: x2 + x – 6 = (x + 3)(x – 2)
Diamond Method Do you recognize the pattern??? Complete the pattern. ___________________________________________________________________________________________________________________________________________________________________
Multiply
(x + 2)(x + 5) = _____________________________ = ___________________
Notice the constant term in the trinomial; it is the product of the constants in the binomials.
You can use this fact to factor a trinomial into its binomial factors.
(Find two factors of c that add up to b)
11
Practice: Factor Completely.
a. x2 + 10x + 21
b. x2 - 13x + 40
c. 2x2 - 4x - 96
Example 2: Factoring Polynomials of the form ax2 + bx + c; trinomials where a > 1
Method 1 Rainbow: Example: Factor: 2x2 – 5x – 3
Step 1: Check for any common factors. (GCF = 1)
Step 2: Multiply the a and the c term of your leftovers. Rewrite the trinomial without the leading
coefficient (a) and with the product as your new c term. Leave the middle term the same.
2x2 – 5x – 3
(a)(c) = (2)(-3) = -6
New trinomial: x2 – 5x – 6
Step 3: Factor your new trinomial.
( x – 6 ) ( x + 1)
Step 4: Divide your inside numbers by the a term (2). Reduce the fraction if possible.
( x – 6 ) ( x + 1)
2 2
( x – 3 ) ( 2x + 1 )
Step 5: Since 2 cannot divide 1 evenly, take
the 2 and put it in front of the x.
12
Practice: Factor completely. 4x2 – 12x + 5
Practice: Factor completely. 3t
3 – 5t
2 – 12t
13
Practice: Factor completely. 6a2 + 10ab – 4b
2
Challenge:
14
Summary: Exit Ticket: Exit Ticket:
15
Day 3: Homework:
Homework: Textbook page 26-27 #15 – 25 (odd), 28 - 40 (even), 42 – 45
Homework Answers
16
Day 4: SWBAT: Review all 4 Factoring Techniques
Polynomial GCF D.O.T.S. Trinomial Grouping
x2 + 15x + 54
12116 2 x
x
3y
4 + x
2y
2
12194 2 aa
17
18
Day 4: Homework
19
Answers
33) 34)
35) 36)
20
Day 5: Simplifying Rational Expressions Warm – Up:
A rational expression is the quotient of two polynomials. Each of the following fractions is a
rational expression:
Division by Zero if not defined. A rational expression has no meaning when the denominator is
zero.
Example 1: Find the value for x that makes the fraction undefined (excluded values):
c. 2x+1
x2- 15xa.
6
x+4b.
3
3x-6
4
3
x
x
2
5
ab
a
4
12
65
22
yy
y
21
a) 2x-10
2x b) 4x-8
4x
Practice: Find the value for x that makes the fraction undefined (excluded values):
A rational expression is in simplest form when its numerator and denominator have no
common factors other than 1 and -1.
*** When simplifying a rational expression you should always FACTOR first and then cancel
out the common factors***
Example 2: Reduce to lowest terms (factoring and cancelling).
Think:
=
To reduce a fraction to lowest terms:
1) Factor both the numerator and the denominator.
2) Divide both the numerator and the denominator by their
greatest common factor by canceling the common factor.
=
22
1) 10x-20
10
a) x+5
x2+4x-5
Practice: Reduce to lowest terms.
2)
Example # 3: Reduce to lowest terms (factoring trinomials and cancelling).
Practice: Reduce to lowest terms.
2) x+1
x2+5x+41)
x+3
x2-x-12
Think:
=
= =
= =
23
Example 4: Reduce to lowest terms (factoring numerator and denominator)
Practice: Reduce to lowest terms
**
a) x2-4
x2-8x+12c)
2x-4
x2+4x-12
1) x2-4
x2+4x+42)
x2-25
x2+2x-153)
5x+5
x2-4x-5
Think:
=
24
Example 6: Recognize Opposites. Simplify if possible and state the excluded values
Case 1:
Case 2:
Case 3:
Note:
__________________________________________________________________________________________
__________________________________________________________________________________________
Example:
Practice:
=
=
25
Challenge Problem:
Summary:
Exit Ticket:
26
Day 5: Homework
Homework: Textbook page 47 – 48 # 6, 7, 8, 10, 14, 15 – 27 odd
__________________________________________________________________________________________
Homework Answers:
Day 6:
27
Day 6: SWBAT Multiply and Divide Rational Expressions
Warm – Up
1. Gfg 2.
=
28
1) 24x
35y14y
8x
Practice: Find the product in lowest terms.
6) 24x3y2
7z321z2
12xy
29
Practice: Find the product in lowest terms.
30
Part 5: Divide Rational Expressions involving Polynomials.
Practice
=
31
c)
d)
32
Challenge Problem: Find the product in lowest terms.
Summary:
Exit Ticket:
33
Day 6: Homework
Homework: Textbook pages 52 – 53 # 3 – 29 odd
Homework Answers:
34
Day 7: ADDING AND SUBTRACTING RATIONALS
SWBAT: To add and subtract rational expressions with the same denominators.
1)
2)
3)
4)
a) 7
12 -
1
12b)
7x
12 -
x
12c)
7
12x -
1
12x
2. Subtract and simplify your answer.
c) 5
6 +
1
6a)
4
9 +
2
9b)
4x
9 +
2x
9
1. Add and simplify your answer.
Warm _Up
35
Example 1: Add the fractions and reduce to lowest terms.
c) 2m + 4
m2 - 9 +
2
m2 - 9
a) 3b
b2 +
5b
b2
36
Practice: Add the fractions and reduce to lowest terms.
1) 2)
3) 4)
Example 2: Subtract the fractions and reduce to lowest terms.
3
6
3
2
xx
x
5a + 2
a2 - 4 -
2a - 4
a2 - 4
b) 3m - 6
m2 + m - 6 -
-m + 2
m2 + m - 6
c)
8
4
8
2 xx
9
3
9 22 xx
x
d
c
d
c
12
9
12
19
10x
5y -
2x
5y
a)
37
Practice: Subtract the fractions and reduce to lowest terms.
7) 8)
9) 10)
11)
+
12)
Challenge Practice:
13)
y
b
y
b
3
4
3
11
62
64
62
48
x
x
x
x
1
65
1
5622 x
x
x
x
yx
yxy
yx
xyx
2
2
2
2 22
38
Summary:
Exit Ticket:
39
Day 7: Add or Subtract. Simply your answer.
1) 2)
3) 4)
5) 6)
7) 8)
9) 10)
22 3
5
3 x
x
x
x
16
4
16 22 xx
x
33
44
y
y
y
y
3
3
3
122
xx
x
33 2
3
2
7
xx
6
6
6
2
x
x
x
x
22
22
22
3 22
x
xx
x
xx
254
8
254
322 x
c
x
x
65
1
65
13422 xxxx
x
23
65
23
2722 xx
x
xx
x
40
Day 8: ADDING AND SUBTRACTING RATIONALS
SWBAT: To add and subtract rational expressions with unlike denominators.
1)
2)
Part 1: Identifying LCM
Find the LCM of the given expressions
A.
a) 7
12 -
1
12b)
7x
12 -
x
12c)
7
12x -
1
12x
2. Subtract and simplify your answer.
c) 5
6 +
1
6a)
4
9 +
2
9b)
4x
9 +
2x
9
1. Add and simplify your answer.
Warm _Up
41
42
Example 3: Subtract
(
)
Example 4: Add
43
Practice: Subtract
Example 5:
Practice:
44
Summary/Closure
Exit Ticket
45
Day 8 : Homework
Homework: Textbook Pages 56 – 57 #’s 3 – 23 odd
Homework Answers
46
Sample Regents Questions:
1)
2)
3)
4)
5)
47
Day 9: Complex Rational Expressions
SWBAT: simplify complex rational expressions
Warm Up:
A complex fraction is a fraction whose numerator, denominator, or both contain
fractions. Some examples of complex fractions are:
A complex rational expression has a rational expression in the numerator, the
denominator, or both. For example, the following are complex rational expressions.
7
2
5
5
13
2
b
bb1
211
2
x
x
1
3
a
a
1
48
Method for simplifying a complex fraction:
Example 1: Simplify
( )
( )
( ) ( )
49
Example 2:
Example 3:
50
Example 4:
Example 5:
Example 6:
2
11
11
a
a
51
Example 7:
Example 8:
32
62xx
xx
2
2
91
341
x
xx
52
Example 9:
Summary/Closure
Exit Ticket:
610
5332
a
aa
53
Day 9: Homework
Homework: Textbook Page 64 #’s 7 – 23 odd
Homework Answers:
54
Day 10: More Practice with Complex Fractions
Answers:
55
Answers:
56
Day 11: Rational Equations
SWBAT: solve rational equations
Warm Up:
Example 1: Solving a Rational Equation
2
7
5
3
10
xx
Step 1: Find the LCD
Step 2: Multiply every term by the LCD
Step 3: Simplify and solve.
57
Example 2:
58
Example 3:
Example 4:
59
Example 5:
Example 6:
*Regents Question*
60
Summary:
Exit Ticket:
61
Day 11: Homework #1 – 47 every other odd
62
Answers:
63
Day 12: More Practice with Rational Equations
64
65
Answers
66
Algebra 2 Trig (Mixed Review)
Perform the indicated operation.
1.
2.
3.
4.
5.
6.
67
7.
8.
a) Determine the value(s) for which the rational expression has no meaning.
b) Simplify:
a)
b)
Ans: a) b)
68
PRACTICE TEST!
1)
2)
3)
4)
5)
69
6)
7)
8)
9)
70
10) Solve for t.
11)
