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

2. 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

3. 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

4. 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
2* *16
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

5. 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

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

7. 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

8. 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 * = 3² * 2² 4. 27= 3³ 5. 54= 3³ * = 2 5 * = = 2² * 5² 9. 75= 3 * 5²
Find the greatest Common Factor of Each Pair:
and 60 = 12
and 49 = 7
and 121= 11

9. 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)

10. 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)

11. 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)

12. 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)

13. 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)

14. 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

15. 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

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

17. 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 Sum
1 and NO
2 and NO = *6= 18
3 and YES!!! (x+3) (x+6)= x² +9x +18

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!!

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

20. 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

21. 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

22. 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
1 * 6
2 * 3
The constant term in the trinomial is the product of the constants in the binomial
1*10
2* 5
See which terms work out to give you… x²+ 19x +10
(3x +2) (2x+5) = 6x²+ 19x +10
ALWAYS FOIL and CHECK YOUR WORK!!!

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

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

25. 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

26. 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

27. 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) (2x+3) (x-4) 10. (2y-7) (y-2) (2n-1)(2n+9) 11. (5x+4)(x+1) (5x +3) (x-2) 12.(x+2)(3x+1) (x-2) (6x-1) 13.(2a-1)(2a+5) 23.-1(2x-1)(2x+5) 14.(3x-1)(5x+3) (5x+9)(x-2) 15.(2x-3)(x+2) 16.(3n+2)(2n-5)

28. 7.6 Choosing a Factoring Method

29. 7.6 Factoring by Multiple Methods
2x²+ 5x+ 4x+ 10
2x²+ 9x+10

30. 7.6 Factoring by Multiple Methods

31. 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

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

33. Review
Chapter 7 Test!!!!
Page , 21-26, 28-32, 33-40

34. Review Page , 21-26, 28-32, 33-40
3. 2² * (b+2)(b+4) 4. 2² * x 32. (x²+7)(x-3) b² 33.(n²+1)(n-4) 6. Prime r 34.(2b+5)(3b-4) 7. 2³ * x(1-3x²) 35.(2h²-7)(h+7) (-b +2) 36.(3t+1)(t+6) 9. 2*3* (2v+3) 10. 2*3* (a²-3a-2) g(g²-3) (g²+1) (4p²-p+3) (2x+9) (x-4) (t-6)(3t +5) (5-3n)(6-n)