Algebra 1 Unit 6B Factoring
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1 Algebra 1 Unit 6B Factoring Monday Tuesday Wednesday Thursday Friday 9 A Day 10 B Day 11 A Day 12 B Day 13 A Day Test Exponents and Polynomials Factor GCF and Trinomials box method Factoring Trinomials Feb B Day 18 A Day 19 B Day 20 A Day No School Staff Development Factoring Trinomials Factoring with Patterns GCF difference of squares perfect square trinomials Dividing Polynomials Retest CBA #4 Quiz Factoring 23 B Day 24 A Day 25 B Day 26 A Day 27 B Day Divide Polynomials Quiz Factoring Elaboration day Test CBA #6 (grade will need to go on the NEXT six weeks marking period) 1
2 WARM-UP # Simplify each expression. 1. (x + 4)(x 6) 2. (10x 2 + 5x 6) (8x 2 2x + 7) 3. 5x 2 (2xy 3x) 4. (2x + 5)(3x + 6) 5. (x 2 + y 2 ) (-x 2 + y 2 ) 6. a b ab
3 Notes GCF and Factoring Prime Number a whole number, greater than 1, whose only factors are 1 and itself Composite Number a whole number, greater than 1, that is not prime Prime Factorization a whole number expressed as a product of factors are all prime numbers (i.e. factor tree) Greatest Common Factor (GCF) the greatest common factor of two or more integers is the greatest number that is a factor of all the integers EX1: State whether each number is prime or composite. If the number if composite, find its prime factorization (tree). a. 28 b. 61 c. 112 d. 150 EX2: Find the GCF between two numbers using the calculator. a. -45, 15 b. 169, 13 c. -20, 440 d. 96, 12, -8 Greatest common factor for the same variable will be LOWEST exponent of that given variable. Factoring to express a polynomial as the product of a monomial and a polynomial EX3: Find the GCF for each set of monomials. a. x 2, x 5, x 4 b. 49x, 343x 2 c. 4a 7 b, 28ab d. 96y, 12x, -8y 3
4 EX4: Factor each polynomial. Notes GCF and Factoring a. 24w + 72z b. 30ab 2 + a 2 b 12ac 3 c. x 4 18x x d. a + 10a 2 b 3 e. 88x 4 11x x 5 f. 14c 3 42c 5 49c 4 g. 48w 2 x + 18wx 2 36wx h. -x 5 4x x 3 x 6 i. 8x 7y + w j. 18y 2 50 k. x 3 + 2x 2 + x 4
5 Reverse Distribution Find a monomial and a trinomial whose product is equal to each problem below. Cut and paste it in the correct place. Problems Monomials (GCF) Trinomials 1. 12x 2 + 3x x 4 6x 2 + 3x 3. 24x x 4 4x x 4 12x 3 + 6x x 3 24x 2 12x 6. 10a 4 b 2 5a 3 b + a 2 b 7. 5a 6 b a 5 b 4 15a 4 b a 5 b a 4 b 4 30a 3 b a 6 b 2 30a 5 b a 4 b a 7 b 6 15a 5 b a 3 5
6 Monomials (GCF) Trinomials 6x (10a 2 b 5a + 1) 5a 4 b 3 (10a 4 b 6 3a 2 b 2 + 5) 10a 4 b 2 (a 2 b 2 + 2ab 3) 5a 3 (4x 2 + x 2) 3 (x 2 4x 2) 3x (2x 2 6x + 3) a 2 b (2a 2 b 2 + ab 3) 10a 3 b 3 (6x 2 + 3x 1) 4x 3 (5a 2 3ab + b 2 ) 2x 2 (4x 3 2x + 1) 6
7 Name Date GCF and Factoring Factor out the GCF. 1. x 3 + x 2 + x 2. 15a + 12b + 6c 3. 8x 2 18y 2 4. x 2 y 2y 5. z 3 + 4z 6. 4x 2 4x 7. 15x 2 50x a 11b 9. 64c 3 56c c x 6 y 3 32x 3 y 2 20x 2 y x x x 3 y 4 40xy 5 7
8 Simplify each expression. 13. (2x + 5xy + 7y) + (3x + 7xy + y) 14. (3x + 2y) (5x + 6y) 40a b 15. (a + b) a b (2m -4 n 3 )(-5mn -7 ) 18. a b Find the volume of a cylinder with a diameter of 4x 3 y and a height of 7x 2 y Find the volume of a cube with sides 2b 3 r 2. Solve x + 2 = 2(5x 11) x < 10 8
9 WARM-UP # Find the missing information on the given rectangles What is the area? This is the same size rectangle just divided up What is the area of the first rectangle? 12 What is the area of the second rectangle? What is the area of the whole rectangle? Write the area of each rectangle inside each box for both of the rectangle below and answer the questions What is the total area of the rectangle? What is the total area of the rectangle? 6 What do you notice about all of the rectangles above? What is special about the length and width? What is the length and width of all the rectangles? 9
10 EXPLORE Given the rectangles below, determine the length and width of each rectangle and the area. Rectangles are not drawn to scale Total Area Total Area 6 Length Width 21 Length Width Total Area Length 2x 5x Total Area Length 21 Width Width 3x 2 2x Total Area Length 4x 2x 1 Total Area Length 30x 20 Width -1 Width x 2 Total Area Length x x 2-4 Total Area Length 30x 20 Width -12 Width Total Area Length (x 1) Width (2x + 3) 10
11 Notes Factoring Trinomials EX1. Recall the box method to multiply two binomials. Multiply (x 3)(x + 2). Factors: Product: EX2. Find the missing dimension of each trinomial s box. Fill in the blank cells in each box. a. a 2 + 7a + 10 = (a + 5)( ) b. c 2 10c + 21 = (c 3)( ) a a 2 c 2 c c. y 2 2y 15 = (y + 3) ( ) d. n 2 + 3n 28 = (n 4) ( ) y y 2 n 2 n e. How do the quantities you filled in the 2 blank cells relate to the original trinomial? 11
12 Notes Factoring Trinomials EX3. Write the numbers that give a sum of 5x and a product of 50x 2. Standard form: EX4. Factor each trinomial. a. x 2 + 7x + 10 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) b. x 2 + 3x 4 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) c. x 2 64 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) 12
13 Notes Factoring Trinomials d. 3x x + 8 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) e. 2y 2 7y + 6 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) f. 6x 2 21x 12 = Sums to be: (Middle term: b) Yield a product of: (this comes from multiplying the a and c) 13
14 Notes Factoring Trinomials Factoring Using Algebra Tiles EX5. Determine the factors of each polynomial. a. b. c. d. 14
15 Name Date A B C D E (x 2)(5x 8) (2x + 3)(3x 2) (x 13)(x + 3) (x + 2)(x 3) (x 2)(x + 3) F G H I J (x 8)(2x + 5) (x + 2)(7x + 3) (x 2)(x + 1) (x 5)(x + 3) (x + 5)(x 3) K L M N O (x 7)(x + 7) (x 2)(x 8) (x + 2)(x + 8) (x 7)(x 2) (x + 7) 2 P Q R S T (x 7)(x + 2) (x + 2)(x 8) (x 3)(x + 3) (x 6 )(2x 1) (x + 6) 2 U V W X Y (x 5)(x + 5) (x 5)(x 5) (x 3) 2 (x 3)(x + 20) (x 6)(x 2) Z (x + 9)(x + 7) Factoring Trinomials Directions: Match that answer to the correct letter of the alphabet. Enter that letter of the alphabet on the blank corresponding to the problem number. Factor each polynomial make sure to show your work. 1. x x x x x x 2 9x x 2 18x x 2 2x 15 15
16 7. x x 2 8x x 2 13x x 2 + x x 2 10x x 2 11x x x x 2 + 5x x 2 x x x x 2 9 Identify the simplified area of each rectangle. Then determine the factors
17 WARM-UP # 1. Factor: x x + 12 Factors: What did you notice? 2. Factor: 4x 2 9 Factors: What did you notice? 3. Factor: x 2 + 6x + 9 Factors: What did you notice? 17
18 18
19 Explain Factoring with Patterns Difference of Squares a 2 b 2 = (a) 2 (b) 2 = (a + b)(a b) difference opposite signs Conjugate pairs *Warning: a 2 + b 2 does not factor To recognize perfect squares, look for coefficients that are squares of integers and variables raised to even powers. EX1: Factor, if possible, using the difference of squares. a. 4x 2 9y 2 b. a 2 16b 2 c. 9x 4 25y 4 d. u 2 v 2 w 2 z 2 e. 25m n 2 19
20 Explain Factoring with Patterns Perfect Square Trinomials a 2 + 2ab + b 2 = (a + b)(a + b) = (a + b) 2 a 2 2ab + b 2 = (a b)(a b) = (a b) 2 EX2: Factor each of the following. a. x 2 + 6x + 9 b. x 2 10x + 25 c. a 2 + 8a + 16 d. 9a 2 24a
21 Name Date Factoring Patterns Determine whether each statement is TRUE. If not, find the correct product. 1. (3x + 1) 2 = 9x 2 + 6x (m 4) 2 = m 2 16m (5t 2) 2 = 25t 2 20t (2n + 7) 2 = 4n n (2b + 3) 2 = 4b b (2a + b) 2 = 4a 2 + 4ab + b 2 Factor each polynomial. If it cannot be factored, write prime. 7. t 2 12t a 2 + 2ab + b t t n n a 2 24ab + 9b 2 21
22 13. t 2 18t n t + t n m 2 16n a 2 8a a a n Which is the correct factorization of 45x y 2? A. 5(3x + 2y) 2 B. 5(3x 2y) 2 C. 5(3x + 2y)(3x 2y) D. 5(3x + 2y)(3x 2y) 22. Challenge Determine the value(s) of k for which each expression is a perfect square trinomial. a. 49x 2 84k + k b. 4x 2 + kx
23 Explore Dividing Polynomials Remember when we MULTIPLIED: (using a box) (x + 2)(x + 6) or (2y + 1)(3y 4) So can you now DIVIDE these polynomials: (using a box) x 2 + 8x + 12 x + 2 6y 2 5y 4 3y 4 3y 4 x +2 So.. x 2 + 8x + 12 x + 2 the quotient is: 6y 2 5y 4 3y 4 the quotient is: 23
24 24
25 Explain Dividing Polynomials Dividing is the opposite operation of. Therefore, we will use the to assist in dividing trinomials when given a trinomial divided by a binomial. EX1. Simplify each expression. a. x 2 + 4x 5 x 1 Quotient: x x 2 5x 1 x 5 b. 2x x + 12 x + 4 Quotient: 2x 2 8x 3x 12 c. 6x x 5 3x 1 Quotient: 25
26 Explain Dividing Polynomials EX2. Simplify each expression ON OUR OWN. a. 5x x 14 x + 7 Quotient: b. x 2 11x + 24 x 8 Quotient: c. 2x x 9 2x 1 Quotient: 26
27 Name Gingerbread Man Date Divide each polynomial. Each answer determines the next location of the traveling gingerbread man. Determine the path the gingerbread man makes through the school. 1. x x + 27 x x 2 13x + 40 x 8 3. x 2 7x 44 x x 2 + x 42 x x 2 + 9x + 4 2x x 2 15x + 7 x x 2 + 7x 15 2x x x + 5 2x x x 14 x x 2 + 7x 12 3x x 2 + 2x 3 2x x 2 15x + 4 3x 4 27
28 Front Door Cafeteria (3x 1) Principal Counselor Nurse (x 2) (4x + 3) (x + 11) Secretary (4x 3) Attendance (x + 1) Trophy case (3x 2) Teacher Workroom K - 2 Kindergarte n 1 st Grade 2 nd Grade (x + 2) (2x 1) (3x + 1) Library 3 rd Grade (x 4) (x + 5) Playground 2 (x + 6) (x 5) 4 th Grade 5 th Grade (3x + 2) (x + 4) Teacher Workroom 3 5 Theater (x + 3) Home Ec Lab Where s my class? Computer Lab (x 11) Art Room (x + 7) Playground 1 (x 7) 28
29 Name Date 1. Find the volume of a cube with sides 2b 3 r 2. Review CBA #6 2. Find the volume of a cylinder that has a radius of 5s 3 t 5 and a height of 2s 2 t Find the area of a triangle that has a base of 32mn 7 and a height of 3m 4 n If a rectangle has an area of 16x 7 y 4 and a length of 4x 3 y, what is its width? 5. Distance (d), rate (r), and time (t) are related by the formula d = rt. If a ball rolls 36p 4 q 9 feet for 4p 2 q 3 minutes, what is the rate? 6. Write an expression that best represents the area of a square with sides of 7x 4 y 3? 7. Find the perimeter and area of the rectangle in terms of n. 3n 5 2n
30 8. Find the perimeter and area of the triangle in terms of x. 3n + 5 5n 1 2n Simplify each expression. 9. (-2x + x 2 ) x(5x 4) + (9x 2 6x) 10. p(2p 3) + (p 3)(4p + 1) 11. (3x 5 ) 3 (2x 7 ) (-3x 6 ) (3r + 7) x y z x y z ( 2a b ) ( 7ab ) a b n 6 + n + n The dimensions of a wall are 7xy feet by 8x 2 y 3 feet. A picture has dimensions 2x feet by x 2 y 4 feet. If the picture is hanging on the wall as shown, what is the area of the wall not covered by the picture? 18. A pitcher contains 16x 5 y 4 ounces of water. A mug holds 2x 2 y ounces. Leticia pours water from the full pitcher into mugs. If she filled ax b y c mugs, what is the value of a + b + c? 30
31 19. Find the area of a circle with radius 6r 3 s 5 inches. 20. Find the area of a rectangle with side lengths (x 2 7x) and (2x 2 + 3x + 1). 21. Describe and correct the error in finding the product of the given polynomials. 22. The area of a rectangle is 3x 2 10x 8. Find the dimensions (length and width) of the rectangle. Factor out the greatest common monomial factor a 2 40b s s 2 54s abc 2 6a 2 c 31
32 Completely factor each of the following polynomials. 26. r 2 + 2r y 2 2y x x ax 2 3ax 35a 30. 4a 2 + 9a k k x k u 2 25 Identify the simplified area of each rectangle. Then determine the factors
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