AT Packet #1: Rational Epressions/Equations Name: Teacher: Pd:
Table of Contents o Day 1: SWBAT: Review Operations with Polynomials Pgs: 1-3 HW: Pages -3 in Packet o Day : SWBAT: Factor using the Greatest Common Factor (G.C.F.) Pgs: 4-8 HW: Tetbook page 6-7 (Page 9 in Packet) #3 8, 9-14, 7, 9, 31, 3, 33, 37, 38 o Day 3: SWBAT: Factor quadratic trinomials of the form a + b + c. Pgs: 10-14 HW: Homework: Tetbook page 6-7 (Page 15 in Packet) #15 5 (odd), 8-40 (even), 4 45 o Day 4: SWBAT: Review of Factoring Pgs: 16-17 HW: Pages 18-19 o Day 5: SWBAT: Simplify Rational Epressions Pgs: 0-5 HW: Homework: Tetbook page 47 48 (Page 6 in Packet) # 6, 7, 8, 10, 14, 15 7 odd o Day 6: SWBAT: Multiply and Divide Rational Epressions Pgs: 7-3 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: 34-38 HW: Page 39 in Packet o Day 8: SWBAT: Adding and Subtracting Rational Epressions with Unlike Denominators Pgs: 40-44 HW: Tetbook Pages 56 57 (Pages 45-46 in Packet) # s 3 3 odd and Page 45 o Day 9-10: SWBAT: Simplify Comple Fractions Day 9: Pgs: 47-5 Day 9: HW: Tetbook Page 64 (Page 53 in Packet) # s 7 3 odd; Day 10: Pages 54-55 o Day 11-1: SWBAT: Solve Rational Equations Day 11: Pgs: 56-60 HW: Pages 61-6 in Packet #1 47 every other odd Day 1: Pages 63-65 o Review Pages 66-67 o Practice Test - Pages 68-70 TEST 1:
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 5 + 9) + ( 3 3 ) = - 3 4 + 9 ***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) = 6 +15 4 10 = 6 + 11-10 1
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) (3 10 + 3) = 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.
Solving Equations and Inequalities 1) 5 + 4 = 39 ) 7a + 3 > 17 3) 7 + 5 = 4 + 3 4) (b 1) (3b 4) = b 5) (b 3) + 3(b + 4) = b + 14 6) -3 1 + 7) 4( + ) (3 + 4) = + 18 8) 5y 1 y + 5 3
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
Practice: Factor each polynomial using the GCF and check your answer. a. 7n 3 + 14n + 1n b. a b 3 + ab c c. 1 7 y 3 15 3 y 6 + 9 4 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
Eample 3: Practice: G H 6
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. ** 7 5 75 3 7
Challenge Problem: Summary: Eit Ticket: 8
Day : Homework: Homework: Tetbook page 6-7 #3 8, 9-14, 7, 9, 31, 3, 33, 37, 38 Homework Answers 9
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: + - 6 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
Practice: Factor Completely. a. + 10 + 1 b. - 13 + 40 c. - 4-96 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
Practice: Factor completely. 4 1 + 5 Practice: Factor completely. 3t 3 5t 1t 1
Practice: Factor completely. 6a + 10ab 4b Challenge: 13
Summary: Eit Ticket: Eit Ticket: 14
Day 3: Homework: Homework: Tetbook page 6-7 #15 5 (odd), 8-40 (even), 4 45 Homework Answers 15
Day 4: SWBAT: Review all 4 Factoring Techniques Polynomial GCF D.O.T.S. Trinomial Grouping + 15 + 54 16 11 3 y 4 + y 4a 19a 1 16
17
Day 4: Homework 33) 34) 18
35) 36) Answers 19
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: 3 4 5 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): 6 3 5 +1 a. b. c. 4 +4 3-6 9-15 5 4 0
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) -10 10 4 3 b) 4-8 4 3 = = 1
Practice: Reduce to lowest terms. 1) 10-0 10 ) + 6 5 + 30 = = Eample # 3: Reduce to lowest terms (factoring trinomials and cancelling). Think: a) +5 +4-5 = Practice: Reduce to lowest terms. 1) +3 --1 ) +1 +5+4 = =
Eample 4: Reduce to lowest terms (factoring numerator and denominator) c) a) -4-4 64 +4-1 -8+1 1 + 3 Think: = Practice: Reduce to lowest terms 1) -4 +4+4 ** 5+5-5 ) + 8 3) +-15-4-5 + 48 3
Eample 6: Recognize Opposites. Simplify if possible and state the ecluded values Case 1: Case : Case 3: + Note: Eample: = Practice: = 4
Challenge Problem: Summary: Eit Ticket: 5
Day 5: Homework Homework: Tetbook page 47 48 # 6, 7, 8, 10, 14, 15 7 odd Homework Answers: Day 6: Day 6: SWBAT Multiply and Divide Rational Epressions 6
Warm Up 1. Gfg. = 7
Practice: Find the product in lowest terms. 1) 4 35y 14y 6) 43 y 8 7z 3 1z 1y 5 1 + 1 + 10 8
Practice: Find the product in lowest terms. 1b 5b + 15 b 9 3b + 9b 5 ( + 3) + 5 + 6 9
Part 5: Divide Rational Epressions involving Polynomials. = Practice 30
c) 3a a+3 5a a+6 3 a d) 31
Challenge Problem: Find the product in lowest terms. Summary: Eit Ticket: Day 6: Homework 3
Homework: Tetbook pages 5 53 # 3 9 odd Homework Answers: 33
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) 4 9 + 9 Warm _Up b) 4 9 + 9 c) 5 6 + 1 6. Subtract and simplify your answer. a) 7 1-1 1 ) 6y 3y 4y 1 3y b) 7 1-1 c) 7 1-1 1 3) 3 7 + 7 4) 3 7 7 34
Eample 1: Add the fractions and reduce to lowest terms. a) 3b b + 5b b + b + + 10 c) m + 4 m - 9 + m - 9 + m 9 b + 10 m 9 35
Practice: Add the fractions and reduce to lowest terms. 1) 4 ) 8 8 19c 1d 9c 1d 3) 3 4) 9 9 6 3 3 Eample : Subtract the fractions and reduce to lowest terms. a) 10 5y - 5y b) 5a + a - 4 - a - 4 c) 3m - 6 a - 4 m + m - 6 - -m + m + m - 6 + ( ) 5y + ( ) a 4 + ( ) m + m 6 5y a 4 m + m 6 a 4 m + m 6 36
Practice: Subtract the fractions and reduce to lowest terms. 7) 11b 4b 8) 3y 3 y 8 4 4 6 6 6 9) 6 5 5 6 10) 1 1 + 14 + 3 + + 3 11) 3y (3y ) + ( y) y y y 1) 3 y+ 3y y+ + +y y+ Challenge Practice: 13) y y y y y 37
Summary: Eit Ticket: 38
39 Day 7: Add or Subtract. Simply your answer. 1) ) 3) 4) 5) 6) 7) 8) 9) 10) 3 5 3 16 4 16 3 4 3 4 y y y y 3 3 3 1 3 3 3 7 6 6 6 3 5 4 8 5 4 3 c 6 5 1 6 5 13 4 3 6 5 3 7
Day 8: ADDING AND SUBTRACTING RATIONALS SWBAT: To add and subtract rational epressions with unlike denominators. 1. Add and simplify your answer. 1) + 1 5 a) 4 9 + 9 7 1 5 Warm _Up b) 4 9 + 9 c) 5 6 + 1 6 ) 1 5 + 1. Subtract and simplify 7 your answer. a) 7 1-1 1 b) 7 1-1 c) 7 1-1 1 Part 1: Identifying LCM Find the LCM of the given epressions A. 4 7, 3 5 B. 3 c + 3, 7 c + 6c + 9 40
3 + 5 = 41
Eample 3: Subtract 5 +3 + 3 +3 5 ( )( ) ( ) ( ) 5 ( )( ) ( )( ) ( )( ) ( )( ) Eample 4: Add 3y+1 y 16 + y y+8 4
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
Eit Ticket 44
Homework: Tetbook Pages 56 57 # s 3 3 odd Day 8 : Homework Homework Answers 45
Sample Regents Questions: 1) ) 3) 4) 5) 46
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: 5 7 3 1 5 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 3 1 1 b b 1 b 1 47
Method for simplifying a comple fraction: or Eample 1: Simplify 16 1 8 1 Method 1: horizontal method Method : Clearing Denominators 48
Eample : Eample 3: 49
Eample 4: Eample 5: Eample 6: 1 1 a 1 1 a 50
Eample 7: 6 3 Eample 8: 4 1 1 3 9 51
Eample 9: 3 5 a a 10 6 a 3 Summary/Closure Eit Ticket: 5
Day 9: Homework Homework: Tetbook Page 64 # s 7 3 odd Homework Answers: Day 10: More Practice with Comple Fractions 53
Answers: 54
Answers: Day 11: Rational Equations 55
SWBAT: solve rational equations Warm Up: 1 4 = 1 + 1 Eample 1: Solving a Rational Equation 10 3 5 7 Step 1: Find the LCD Step : Multiply every term by the LCD ( ) 10 + ( ) 3 5 = ( ) 7 Step 3: Simplify and solve. Eample : 1 + 1 = 3 1 3 56
57
Eample 3: + = 3 + 4 (+) Eample 4: y+1 3y 18 5 y 6 = 1 3 58
Eample 5: a a = a+ a Eample 6: *Regents Question* 59
Summary: Eit Ticket: 60
#1 47 every other odd Day 11: Homework Answers: 61
Day 1: More Practice with Rational Equations 6
63
Answers 64
Algebra Trig (Mied Review) Perform the indicated operation. 1.. 3. 4. 5. 6. 65
7. 8. 6(b 5) b a) Determine the value(s) for which the rational epression has no meaning. b) Simplify: a) b) Ans: a) b) + 5 66
PRACTICE TEST! 1) ) 3) 4) 5) 67
6) 7) 8) 9) 68
10) Solve for t. 11) 69