chapter 7 factors and greatest common factors ms. fisher

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


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 Tree12 322*6 2*16 2*3 2*4 2*2

1.Align common factors2.Find the product of common factors12 2*232 2*2

The greatest of the common factors is 4


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 TimePage 459

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

7.2 Factoring by GCFObjective: Factor polynomials by using the greatest common factor

Warm up: Who can explain the Distributive Property?

a(b+c)= ab +acAgenda: Go over HW problems from 7.1

Whole group- Teach section 7.2



page 467

7.2 Factoring by GCFHomework 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= * 3

7. 7= 7

8. 100 = 2² * 5²

9. 75= 3 * 5²

Find the greatest Common Factor of Each Pair:

10. 12 and 60 = 1211. 14 and 49 = 712. 55 and 121= 11

7.2 Factoring by GCF1. 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 GCF1. 10y³+ 20y²-5y

10y³= 2 *5*y*y*y

20y²= 2*2*5*y*y

5y= 5*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

1. 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 1. 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 +cObjective: 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




page 476

7.2 HW page 467 #’S 1-10 AND 12-20

1. 5a(3-a) 12. (5-m)(m-2)

2. g(10g²-3) 13. (2b+5) (b+3)

3. 7(-5x+6) 14. can not be factored

4. -2x(2x+3) 15. (x²+2) (x+4)

5. 2h(6h³ + 4h -3) 16. (2x² +1) (3x +2)

6. 3(x²-3x +1) 17. (2b² +5) (2b-3)

7. m(9m+1) 18. 2(m+2)(m²+3)

8. 7n(2n² + 1 + n) 19. (7r² +6) (r-5)

9. 3(12f+ 6f² +1) 20. (2a²+1) (5a +2)

10.b(-15b+7)

7.3 Factoring x² +bx +cIn 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 +cHow 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!!


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 +cObjective: 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




page 476

7.3 HW page 476 #’s 4-154.(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)

7.4 Factoring ax² +bx +cExample: 6x²+ 19x +10

1. The coefficient of the X² term is the product of the coefficients of the X terms 6

1 * 6

2 * 3

2. 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!!!

7.4 Factoring ax² +bx +c

**** Be careful of your MIDDLE TERM****

Example: 5x² - 14x +8

1. The coefficient of the X² term is the product of the coefficients of the X terms 5

1 * 5

2. 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!!!


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 polynomialObjective: 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)



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+ 102x²+ 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

1. Yes 9. 2p(2q + 1)²

2. No; 2x (4x²-3x-8) 10. 2r (3s+1)(3s-1)

3. Yes 11. mn (n² +m)(n²-m)

4. Yes 12. 2y(x-5)²

5. No; 4 (2p²+1)(2p²-1) 13. 3x²(2x-3)(x+1)

6. Yes 14. unfactorable

7. 3x³(x+2) (x-2) 15. (p³ +1)(p²+3)

8. 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. 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. 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)

chapter 7 factors and greatest common factors ms. fisher

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