A2T. Rational Expressions/Equations. Name: Teacher: Pd:


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1 AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd:
2 Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 13 HW: Pages 3 in Packet o Day : SWBAT: Factor using the Greatest Common Factor (G.C.F.) Pgs: 48 HW: Tetbook page 67 (Page 9 in Packet) #3 8, 914, 7, 9, 31, 3, 33, 37, 38 o Day 3: SWBAT: Factor quadratic trinomials of the form a + b + c. Pgs: HW: Homework: Tetbook page 67 (Page 15 in Packet) #15 5 (odd), 840 (even), 4 45 o Day 4: SWBAT: Review of Factoring Pgs: HW: Pages o Day 5: SWBAT: Simplify Rational Epressions Pgs: 05 HW: Homework: Tetbook page (Page 6 in Packet) # 6, 7, 8, 10, 14, 15 7 odd o Day 6: SWBAT: Multiply and Divide Rational Epressions Pgs: 73 HW: Tetbook pages 5 53 (Page 33 in Packet) # 3 9 odd o Day 7: SWBAT: Adding and Subtracting Rational Epressions with Like Denominators Pgs: HW: Page 39 in Packet o Day 8: SWBAT: Adding and Subtracting Rational Epressions with Unlike Denominators Pgs: HW: Tetbook Pages (Pages in Packet) # s 3 3 odd and Page 45 o Day 910: SWBAT: Simplify Comple Fractions Day 9: Pgs: 475 Day 9: HW: Tetbook Page 64 (Page 53 in Packet) # s 7 3 odd; Day 10: Pages o Day 111: SWBAT: Solve Rational Equations Day 11: Pgs: HW: Pages 616 in Packet #1 47 every other odd Day 1: Pages o Review Pages o Practice Test  Pages TEST 1:
3 Day 1: Operations with Polynomials A monomial is a constant, a variable, or the product of constants and variables. E. 3, a, ab, a 3a 4 ; 3 is the coefficient, a is the base and 4 is the eponent. A polynomial is the sum of monomials. Each monomial is a term of the polynomial. E. 3a + 7a  Adding and Subtracting polynomials When a **When adding and subtracting polynomials, add or subtract the coefficients of like terms. E. 3a + 5a = 8a ( ) + ( 3 3 ) = ***Remember, when you subtract you must change the signs! E. Subtract (3b 4 + b + 3) from (b 4 5b +3). (b 4 5b +3)  (3b 4 + b + 3)= (b 4 5b +3)+ (3b 4 b 3)= b 4 6b Multiplying Monomials and Polynomials **When multiplying monomials, multiply the coefficients and add the eponents of like bases. E. (3a b)(abc) = 6a 3 b c E. ab(a + ab + b ) = a 3 b + a b + ab 3 E. ( 3 y) = ( 3 y)( 3 y) = 4 6 y E. FOIL!!!! (3)(+5) = =
4 HW #1: Operations with Polynomials: Write your answers in simplest form. 1) (3y 5) + (y  8) = ) ( + 3 ) + (4 + 3) = 3) (7b b + 3) (3b + 8b + 3) = 4) (4 3 5) ( ) = 5) a 5 b (7a 3 b ) = 6) (6y ) = 7) y(y y ) = 8) ( + 3)( 1) = 9) (a + 3) = 10) ( + 3)( + 5) = 11) a 3 (a + 3) (a 5 + 3a 3 ) = 1) The length of a rectangle is 4 more than twice the width,. Epress the area of the rectangle in terms of.
5 Solving Equations and Inequalities 1) = 39 ) 7a + 3 > 17 3) = ) (b 1) (3b 4) = b 5) (b 3) + 3(b + 4) = b ) ) 4( + ) (3 + 4) = ) 5y 1 y + 5 3
6 SWBAT: Factor polynomials by using the GCF. Warm Up Day : Factoring by GCF 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 : Find variables that appear in ALL terms and pull out the smallest eponent 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. Eample 1: Factor using GCF: 1 4 y 3 4 y 5 + 8y 4
7 Practice: Factor each polynomial using the GCF and check your answer. a. 7n n + 1n b. a b 3 + ab c c. 1 7 y y y 5 Eample : Factoring a common binomial factor Using the GCF ) 4( + 1) + 7( + 1) 3) y(y )  (y ) 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) 5
8 Eample 3: Practice: G H 6
9 Difference of Two Squares (DOTS) o Binomial o Both terms are perfect squares o Even eponents on variables Divide eponent by o Perfect squares for coefficients Square root coefficient o Pattern: y = ( + y)( y) Eample 4: Factor the binomial below. a 11 Practice: Factor each of the binomials below. I. 4 5y J. 16 9n 8 K. ** 36a 4 b 4 L. **
10 Challenge Problem: Summary: Eit Ticket: 8
11 Day : Homework: Homework: Tetbook page 67 #3 8, 914, 7, 9, 31, 3, 33, 37, 38 Homework Answers 9
12 Warm Up Day 3: SWBAT: Factor quadratic trinomials of the form a + b + c. Trinomials o Has three terms a + b + 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 Eample 1: Factoring Polynomials of the form a + b + c 1) trinomials where a = 1 Eample: Split the : ( )( ) Look for two numbers that multiply to 6 (c) and add to +1 (b): +3,  Final answer: + 6 = ( + 3)( ) Diamond Method Do you recognize the pattern??? Complete the pattern. Multiply ( + )( + 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) 10
13 Practice: Factor Completely. a b c Eample : Factoring Polynomials of the form a + b + c; trinomials where a > 1 Method 1 Rainbow: Eample: Factor: 5 3 Step 1: Check for any common factors. (GCF = 1) Step : 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. 5 3 (a)(c) = ()(3) = 6 New trinomial: 5 6 ( 6)( + 1) Step 3: Place the a term in the first position and the factors in the second. ( 6 ) ( + 1) Step 4: Reduce the factored terms whenever possible. ( 6 ) ( + 1) ( 3 ) ( + 1 ) 11
14 Practice: Factor completely Practice: Factor completely. 3t 3 5t 1t 1
15 Practice: Factor completely. 6a + 10ab 4b Challenge: 13
16 Summary: Eit Ticket: Eit Ticket: 14
17 Day 3: Homework: Homework: Tetbook page 67 #15 5 (odd), 840 (even), 4 45 Homework Answers 15
18 Day 4: SWBAT: Review all 4 Factoring Techniques Polynomial GCF D.O.T.S. Trinomial Grouping y 4 + y 4a 19a 1 16
19 17
20 Day 4: Homework 33) 34) 18
21 35) 36) Answers 19
22 Day 5: Simplifying Rational Epressions Warm Up: A rational epression is the quotient of two polynomials. Each of the following fractions is a rational epression: a 1 4ab y y 5y 6 Division by Zero if not defined. A rational epression has no meaning when the denominator is zero. Eample 1: Find the value for that makes the fraction undefined (ecluded values): a. b. c
23 Practice: Find the value for that makes the fraction undefined (ecluded values): 6 16 A rational epression is in simplest form when its numerator and denominator have no common factors other than 1 and 1. *** When simplifying a rational epression you should always FACTOR first and then cancel out the common factors*** To reduce a fraction to lowest terms: 1) Factor both the numerator and the denominator. ) Divide both the numerator and the denominator by their greatest common factor by canceling the common factor. Eample : Reduce to lowest terms (factoring and cancelling). Think: a) b) = = 1
24 Practice: Reduce to lowest terms. 1) ) = = Eample # 3: Reduce to lowest terms (factoring trinomials and cancelling). Think: a) = Practice: Reduce to lowest terms. 1) ) = =
25 Eample 4: Reduce to lowest terms (factoring numerator and denominator) c) a) Think: = Practice: Reduce to lowest terms 1) ** ) + 8 3)
26 Eample 6: Recognize Opposites. Simplify if possible and state the ecluded values Case 1: Case : Case 3: + Note: Eample: = Practice: = 4
27 Challenge Problem: Summary: Eit Ticket: 5
28 Day 5: Homework Homework: Tetbook page # 6, 7, 8, 10, 14, 15 7 odd Homework Answers: Day 6: Day 6: SWBAT Multiply and Divide Rational Epressions 6
29 Warm Up 1. Gfg. = 7
30 Practice: Find the product in lowest terms. 1) 4 35y 14y 6) 43 y 8 7z 3 1z 1y
31 Practice: Find the product in lowest terms. 1b 5b + 15 b 9 3b + 9b 5 ( + 3)
32 Part 5: Divide Rational Epressions involving Polynomials. = Practice 30
33 c) 3a a+3 5a a+6 3 a d) 31
34 Challenge Problem: Find the product in lowest terms. Summary: Eit Ticket: Day 6: Homework 3
35 Homework: Tetbook pages 5 53 # 3 9 odd Homework Answers: 33
36 Day 7: ADDING AND SUBTRACTING RATIONALS SWBAT: To add and subtract rational epressions with the same denominators. 1. Add and simplify your answer. 1) a 100 a 3a a 0 a) Warm _Up b) c) Subtract and simplify your answer. a) ) 6y 3y 4y 1 3y b) c) ) )
37 Eample 1: Add the fractions and reduce to lowest terms. a) 3b b + 5b b + b c) m + 4 m m m 9 b + 10 m 9 35
38 Practice: Add the fractions and reduce to lowest terms. 1) 4 ) c 1d 9c 1d 3) 3 4) Eample : Subtract the fractions and reduce to lowest terms. a) 10 5y  5y b) 5a + a a  4 c) 3m  6 a  4 m + m m + m + m ( ) 5y + ( ) a 4 + ( ) m + m 6 5y a 4 m + m 6 a 4 m + m 6 36
39 Practice: Subtract the fractions and reduce to lowest terms. 7) 11b 4b 8) 3y 3 y ) ) ) 3y (3y ) + ( y) y y y 1) 3 y+ 3y y+ + +y y+ Challenge Practice: 13) y y y y y 37
40 Summary: Eit Ticket: 38
41 39 Day 7: Add or Subtract. Simply your answer. 1) ) 3) 4) 5) 6) 7) 8) 9) 10) y y y y c
42 Day 8: ADDING AND SUBTRACTING RATIONALS SWBAT: To add and subtract rational epressions with unlike denominators. 1. Add and simplify your answer. 1) a) Warm _Up b) c) ) Subtract and simplify 7 your answer. a) b) c) Part 1: Identifying LCM Find the LCM of the given epressions A. 4 7, 3 5 B. 3 c + 3, 7 c + 6c
43 3 + 5 = 41
44 Eample 3: Subtract ( )( ) ( ) ( ) 5 ( )( ) ( )( ) ( )( ) ( )( ) Eample 4: Add 3y+1 y 16 + y y+8 4
45 Practice: Subtract 7 6 y 49 y y 35 Eample 5: y + 10 y 5 5 y Practice: b 1 b 1 b Summary/Closure 43
46 Eit Ticket 44
47 Homework: Tetbook Pages # s 3 3 odd Day 8 : Homework Homework Answers 45
48 Sample Regents Questions: 1) ) 3) 4) 5) 46
49 Day 9: Comple Rational Epressions SWBAT: simplify comple rational epressions Warm Up: A comple fraction is a fraction whose numerator, denominator, or both contain fractions. Some eamples of comple fractions are: A comple rational epression has a rational epression in the numerator, the denominator, or both. For eample, the following are comple rational epressions. 1 a a b b 1 b 1 47
50 Method for simplifying a comple fraction: or Eample 1: Simplify Method 1: horizontal method Method : Clearing Denominators 48
51 Eample : Eample 3: 49
52 Eample 4: Eample 5: Eample 6: 1 1 a 1 1 a 50
53 Eample 7: 6 3 Eample 8:
54 Eample 9: 3 5 a a 10 6 a 3 Summary/Closure Eit Ticket: 5
55 Day 9: Homework Homework: Tetbook Page 64 # s 7 3 odd Homework Answers: Day 10: More Practice with Comple Fractions 53
56 Answers: 54
57 Answers: Day 11: Rational Equations 55
58 SWBAT: solve rational equations Warm Up: 1 4 = Eample 1: Solving a Rational Equation Step 1: Find the LCD Step : Multiply every term by the LCD ( ) 10 + ( ) 3 5 = ( ) 7 Step 3: Simplify and solve. Eample : =
59 57
60 Eample 3: + = (+) Eample 4: y+1 3y 18 5 y 6 =
61 Eample 5: a a = a+ a Eample 6: *Regents Question* 59
62 Summary: Eit Ticket: 60
63 #1 47 every other odd Day 11: Homework Answers: 61
64 Day 1: More Practice with Rational Equations 6
65 63
66 Answers 64
67 Algebra Trig (Mied Review) Perform the indicated operation
68 (b 5) b a) Determine the value(s) for which the rational epression has no meaning. b) Simplify: a) b) Ans: a) b)
69 PRACTICE TEST! 1) ) 3) 4) 5) 67
70 6) 7) 8) 9) 68
71 10) Solve for t. 11) 69
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