Factors and Greatest Common Factors

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Presentation transcript:

Factors and Greatest Common Factors Chapter 7 Factors and Greatest Common Factors Ms. Fisher

7.1 Factors and Greatest Common Factors Objective: Write the prime factorization of numbers Warm up: What does prime mean? A prime number is a whole number that has exactly two positive factors:________________ Itself and 1 Agenda: Whole group- Teach section 7.1 Small group- Work problems with Ms. Fisher Alex- Differentiation- Lesson Extension page 459

7.1 Factors and Greatest Common Factors Factors: Numbers that are multiplied to find a product. A number is divisible by its factors. Example: Factors of 12 are 1, 2, 3, 4, 6 Prime Number: is a whole number that has exactly two positive factors, itself and 1. The number 1 is not prime because it only has one positive factor. Write the Prime Factorization of 60 Factor Tree 60 2 * 30 10 * 3 2 * 5 The prime factorization of 60 is: 2² * 3 * 5

7.1 Factors and Greatest Common Factors Common Factors: factors that are shared by two or more whole numbers Greatest Common Factor; GCF: the greatest common factor shared between two or more whole numbers. Example: Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 32: 1, 2, 4,8, 16, 32 Common factor: 1, 2, 4 The greatest of the common factors: 4 Factor Tree 12 32 2*6 2*16 2*3 2*4 2*2 1.Align common factors 2.Find the product of common factors 12 2*2 32 2*2 The greatest of the common factors is 4

7.1 Factors and Greatest Common Factors Method #2: Find the greatest common factor 3x³ and 6x² 3x³= 3*x*x*x 6x²= 2*3*x*x GCF= 3*x*x = 3x²

Independent Work Time Page 459 Small group with Ms. Fisher #’s 2-12 Alex: Extension to lesson #’s 2-15 and 17-30

7.2 Factoring by GCF Objective: Factor polynomials by using the greatest common factor Warm up: Who can explain the Distributive Property? a(b+c)= ab +ac Agenda: Go over HW problems from 7.1 Whole group- Teach section 7.2 Small group- Work problems with Ms. Fisher Alex- Differentiation- Lesson Extension page 467

7.2 Factoring by GCF Find the greatest Common Factor of Each Pair: Homework 7.1 Page 459 #’s 2-12 Write the prime factorization of each number 2. 20= 2² * 5 3. 36= 3² * 2² 4. 27= 3³ 5. 54= 3³ * 2 6. 96= 2 5 * 3 7. 7= 7 8. 100 = 2² * 5² 9. 75= 3 * 5² Find the greatest Common Factor of Each Pair: 10. 12 and 60 = 12 11. 14 and 49 = 7 12. 55 and 121= 11

7.2 Factoring by GCF 4x² -3x Step #1: Find the GCF of each term  Ask yourself,what do they have in common? What do they both have? 4x²= 2 * 2* x *x 3 *x= 3 * x The GCF of 4x² and 3x is x Step #2: Use the Distributive Property to factor out the GCF x(4x-3)

7.2 Factoring by GCF 10y³+ 20y²-5y 10y³= 2 *5*y*y*y 20y²= 2*2*5*y*y The GCF of 10y³, 20y², and 5y is 5y. 5y(2y² + 4y -1) (Use the Distributive Property to factor out the GCF)

7.2 Factoring Out a Common Binomial Factor 1. 7(x-3) -2x(x-3) 7(x-3)-2x(x-3) (x-3) is a common binomial factor (x-3) (7-2x) Factor out (x-3) 2. –t(t²+4) + (t² +4) -t(t²+4) + (t²+4) (t²+4) is a common binomial factor -t(t²+4) + 1(t²+4) (t²+4)(-t+1) Factor out (t²+4)

7.2 Factoring by Grouping 12a³-9a²+20a-15 (12a³-9a²) + (20a -15) Group terms that have a common number 3a²(4a-3) + 5(4a-3) Factor out GCF 3a²(4a-3) + 5(4a-3) (4a-3) find common binomial factor (4a-3)(3a²+5)

7.2 Factoring by Grouping with Opposites 3x³-15x²+10-2x (3x³-15x²) + (10-2x) Group Terms 3x²(x-5) + 2(5-x) Factor out the GCF 3x²(x-5) +2(-1) (x-5) Find common binomial factor 3x²(x-5) -2(x-5) Simplify (x-5)(3x²-2)

Independent Work Time Small group with Ms. Fisher page 467 #’s 1-10 and 12-20 Alex Extension page 467 #’s 1-10 and 12-35

7.3 Factoring x² +bx +c Warm up: What is a Quadratic trinomial? CH6 Objective: Factor Quadratic trinomials Warm up: What is a Quadratic trinomial? CH6 Exponent to the 2nd power Three terms x² +bx +c Agenda: Go over HW pg 467 Whole group- Teach section 7.3 Small group- Work problems with Ms. Fisher Alex- Differentiation- Lesson Extension page 476

7.2 HW page 467 #’S 1-10 AND 12-20 5a(3-a) 12. (5-m)(m-2) g(10g²-3) 13. (2b+5) (b+3) 7(-5x+6) 14. can not be factored -2x(2x+3) 15. (x²+2) (x+4) 2h(6h³ + 4h -3) 16. (2x² +1) (3x +2) 3(x²-3x +1) 17. (2b² +5) (2b-3) m(9m+1) 18. 2(m+2)(m²+3) 7n(2n² + 1 + n) 19. (7r² +6) (r-5) 3(12f+ 6f² +1) 20. (2a²+1) (5a +2) b(-15b+7)

7.3 Factoring x² +bx +c In this lesson you will learn how to factor a trinomial into two binomials x² +bx +c Remember the definition of factor: Factors: Numbers that are multiplied to find a product. A number is divisible by its factors. Example: Factors of 12 are 1, 2, 3, 4, 6 Find two FACTORS of c whose sum is b If no such integers exist, the trinomial is not factorable! Example: x² +9x +18 Factors of 18 Sum 1 and 18 19 NO 2 and 9 11 NO 3+6= 9 3*6= 18 3 and 6 9 YES!!! (x+3) (x+6)= x² +9x +18

7.3 Factoring x² +bx +c How do you check your answer? Use the FOIL method!! (x+3) (x+6) should equal x² +9x +18 X * X = X² X * 6 = 6X 3 * X = 3X 3 * 6 = 18 X² +9X +18 YES!! Works!!

Independent Work Time Small group with Ms. Fisher page 476 #’s 4-15 Alex Extension page 476 #’s 4-15 and 20-31

7.4 Factoring ax² +bx +c Objective: Factor Quadratic trinomials of the form ax² +bx +c Warm up: What is different with this Quadratic trinomial then the one we learned yesterday? x² +bx +c Yesterday ax² +bx +c Today No longer have a coefficient of one Agenda: Go over HW pg 476 Whole group- Teach section 7.4 Small group- Work problems with Ms. Fisher Alex- Differentiation- Lesson Extension page 476

4. (x+1) (x+3) 5. (x+2) (x+8) 6. (x+4)(x+11) 7. (x-1)(x-6) 8 4.(x+1) (x+3) 5. (x+2) (x+8) 6. (x+4)(x+11) 7. (x-1)(x-6) 8.(x-2)(x-7) 9.(x-3)(x-8) 10.(x-7)(x+1) 11.(x+9)(x-3) 12.(x+6)(x-5) 13.(x-2)(x+1) 14.(x-6)(x+3) 15.(x-9)(x+5) 7.3 HW page 476 #’s 4-15

7.4 Factoring ax² +bx +c Example: 6x²+ 19x +10 The coefficient of the X² term is the product of the coefficients of the X terms 6 1 * 6 2 * 3 The constant term in the trinomial is the product of the constants in the binomial 10 1*10 2* 5 See which terms work out to give you… 6x²+ 19x +10 (3x +2) (2x+5) = 6x²+ 19x +10 ALWAYS FOIL and CHECK YOUR WORK!!!

**** Be careful of your MIDDLE TERM**** 7.4 Factoring ax² +bx +c **** Be careful of your MIDDLE TERM**** Example: 5x² - 14x +8 The coefficient of the X² term is the product of the coefficients of the X terms 5 1 * 5 The constant term in the trinomial is the product of the constants in the binomial 8 - 1* -8 -2* -4 See which terms work out to give you… 5x²+ 14x +8 (x -2) (5x-4) = 5x²-4x-10x+8 = 5x²-14x +8 ALWAYS FOIL and CHECK YOUR WORK!!!

Independent Work Time Small group with Ms. Fisher page 484 #’s 7-24 Alex Extension page 484 #’s 7-24 and 34-51

Comprehension Check Point Objective: Assess students’ ability to apply concepts Agenda: All students grab a textbook Open to page 489 As a class work through Quiz on Lessons 7.1-7.4

7.6 Combine Methods for factoring a polynomial Objective: To be able to successfully choose an appropriate method for factoring each polynomial. Warm up: Is the below expression completely factored? (2x+6) (x+5) NO! 2(x+3) (x+5) Agenda: Go over HW pg 484 Whole group- Teach section 7.6 (Skip 7.5) Small group- Work problems with Ms. Fisher Alex- Differentiation- Lesson Extension page 501

7.4 HW page 484 #’s 7-24 7. (x+2) (5x +1) 17. (10x+1)(x-1) 8. (x+5) (2x+1) 18. (x+1) (7x-10) 9. (4x-5) (x-1) 19. -1(2x+3) (x-4) 10. (2y-7) (y-2) 20. -1 (2n-1)(2n+9) 11. (5x+4)(x+1) 21. -1(5x +3) (x-2) 12.(x+2)(3x+1) 22. -1(x-2) (6x-1) 13.(2a-1)(2a+5) 23.-1(2x-1)(2x+5) 14.(3x-1)(5x+3) 24. -1(5x+9)(x-2) 15.(2x-3)(x+2) 16.(3n+2)(2n-5)

7.6 Choosing a Factoring Method

7.6 Factoring by Multiple Methods 2x²+ 5x+ 4x+ 10 2x²+ 9x+10

7.6 Factoring by Multiple Methods

Group Work Time Small group with Ms. Fisher page 501 #’s 1-6, 7-16 Alex Extension page 501 #’s 1-6, 7-16, 25-33

HW page 501 #’s 1-6 and 7-16 Yes 9. 2p(2q + 1)² No; 2x (4x²-3x-8) 10. 2r (3s+1)(3s-1) Yes 11. mn (n² +m)(n²-m) Yes 12. 2y(x-5)² No; 4 (2p²+1)(2p²-1) 13. 3x²(2x-3)(x+1) Yes 14. unfactorable 3x³(x+2) (x-2) 15. (p³ +1)(p²+3) 4x(x+1)² 16. 7x³ (x +4) (x-1)

Review Chapter 7 Test!!!! Page 506 3-19, 21-26, 28-32, 33-40

Review Page 506 3-19, 21-26, 28-32, 33-40 3. 2² * 3 16. 3 31. (b+2)(b+4) 4. 2² * 5 17. 2x 32. (x²+7)(x-3) 5. 2 5 18. 9b² 33.(n²+1)(n-4) 6. Prime 19. 25r 34.(2b+5)(3b-4) 7. 2³ * 5 21. 5x(1-3x²) 35.(2h²-7)(h+7) 8. 2 6 22. 16(-b +2) 36.(3t+1)(t+6) 9. 2*3*11 23. -7 (2v+3) 10. 2*3*19 24. 4(a²-3a-2) 11. 5 25. 5g(g²-3) (g²+1) 12. 12 26. 10(4p²-p+3) 13. 1 28. (2x+9) (x-4) 14. 27 29. (t-6)(3t +5) 15. 4 30. (5-3n)(6-n)