Algebra. Chapter 8: Factoring Polynomials. Name: Teacher: Pd:

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1 Algebra Chapter 8: Factoring Polynomials Name: Teacher: Pd:

2 Table of Contents o Day 1: SWBAT: Factor polynomials by using the GCF. Pgs: 1-6 HW: Pages 7-8 o Day 2: SWBAT: Factor quadratic trinomials of the form x 2 + bx + c. (a = 1) Pgs: 9-13 HW: Page 14 o Day 3: SWBAT: Factor a Difference of Two Squares. D.O.T.S Pgs: HW: Page o Day 4: SWBAT: Factor a Polynomial Completely Pgs: HW: Pages o Day 5: SWBAT: More Factoring a Polynomial Completely Pgs: HW: Page 30 o Day 6-7: SWBAT: Stations Review of Factoring Pgs: HW: Pages 34-37

3 Day 1: Factoring by GCF SWBAT: Factor polynomials by using the GCF. Warm Up Multiply each polynomial x(3 2x + 1) 4xy (3x + 6y - 7) Recall! The Property: ab + ac = a(b + c). - 1

4 Steps to Factoring by GCF Step 1: Find largest number that divides into ALL terms. Step 2: Find variables that appear in ALL terms and pull out the smallest exponent 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. Example 1: Factoring by Using the GCF Factor each polynomial and check your answer. a) 2x b) 6x 2-9x Practice: Factor each polynomial using the GCF and check your answer. 1. 7x c c 3. 44n n x 5 18x 5. 10g 3 30g 6. 9m m 2

5 Example 2: Factoring by Using the GCF Factor each polynomial using the GCF and check your answer. c) 7n n + 21n 2 d) 8x 4 + 4x 3 2x 2 Practice: Factor each polynomial using the GCF and check your answer h h 2 6h 8. 36f + 18f n n 4 24n Example 3: Factoring a common binomial factor Using the GCF e) 4x(x + 1) + 7(x + 1) f) y(y 2) - (y 2) Practice: Factor each polynomial and check your answer. 10) 11) 12) 3

6 Example 4: Factor by Grouping g) 4

7 13) 14) 15) 16) 5

8 Challenge Problem: Factor. 12a 2 bc 2-24a 4 c Summary: Exit Ticket: 1) 2) 6

9 Day 1: Homework: Factor using the GCF. 1) x 2 x 2) 6x 2 27x 3) 4x 2 10x 4) 25x 2 10x 5) 5x 2 10x 25 6) 8x 2 4x ) 2x 10x 20x ) 8x 4x 16x 9) 15x 30x 45x ) 4xy 2x y 11) 45 xy 9x y 12) 3x 2 6x 7

10 13) 12x 2 3xy 14) 3xy 2 66y 15) 6 ab 42a ) 18x 9x 17) 4x 3 2 8x 18) x 2x 2 19) 7 xy 21x y 20) ) 2 k t 4k t 22) 23) 24) 8

11 Day 2: Factoring x 2 + bx + c SWBAT: Factor quadratic trinomials of the form x 2 + bx + c. Warm Up 1. Factor by Grouping. 2. Factor using GCF. Mini-Lesson Do you recognize the pattern??? You Try!!! Complete the Diamond Multiply (x + 2)(x + 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) 9

12 ax 2 + bx + c Example 1: First Sign is Positive and Last Sign is Positive Factor: x 2 + 6x + 8 Factor: x 2 + 5x Answer: ( ) ( ) Answer: ( ) ( ) Practice 1: Factor. 1. x 2 + 5x x 2 + 8x x 2 + 6x + 5 Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( ) 4. x 2 + 6x x x x x 10

13 Example 2: First Sign is Negative and Last Sign is Positive Factor: x 2-10x + 24 Factor: x 2-7x Answer: ( ) ( ) Answer: ( ) ( ) Practice 2: Factor. 7. x 2-8x x 2-6x x 2-7x + 10 Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( ) 10. x 2-5x x 2-13x x 2-6x Example 3: First Sign is Positive or Negative and Last Sign is Negative Factor: x 2 + x - 20 Answer: ( ) ( ) 11

14 Practice 3: Factor. 13. x 2 + 2x x 2 + 3x x 2 + 6x - 40 Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( ) 16. x 2-2x x 2-2x x 2-2x - 48 Challenge Problem: 1) 2)Factor: x x

15 Summary: Example: Factor: x 2 5x - 50 Exit Ticket: 13

16 Factor each trinomial. Day 2: Homework: + 13x - 9x - 12x 14

17 SWBAT : Factor a Difference of Two Squares Day 3: Factoring Special Products Warm Up The area of the rectangle below is represented by the polynomial x 2 + 8x + 7. Find the binomials that could represent the lengths and width of the rectangle. A = x 2 + 8x + 7 Make a list of perfect squares. 15

18 Example 1: Factoring the Difference of Two Squares Factor: x 2-25 Practice: Factoring the Difference of Two Squares Factor. 1) x ) x 2-9 3) x ) x ) 49 - x 2 6) x ) x 2 1 8) 4 - x 2 9) x Example 2: Factoring the Difference of Two Squares Factor: 64x 2 1 Factor: x 6-25 Practice: Factoring the Difference of Two Squares 10) 9x ) 9-16x 2 12) 49x ) 25x ) x 2-25y 2 15) 16x 2 25y 2 16) 64x 2 9y 2 17) x 4 y 10 18) 49x 2 121y 2 16

19 Challenge Problem Factor: 1 4 x2-1 9 Summary Exit Ticket: 17

20 Day 3: Homework - Factoring the Difference of Two Perfect Squares 1. x x x x x x x 2 y x 2 25b 2 9. x x x x x x x

21 Factor by Grouping

22 Day 4: Factoring Completely SWBAT: Factor a Trinomial Completely Warm Up Factor each Factor by Grouping. 20

23 Factoring Trinomials Completely In the previous lesson, we saw how to factor a trinomial of the form bx c by employing the diamond method. In each of those cases, the coefficient of the quadratic ( ) term was always one, and thus not written. It is also possible to factor trinomials of the form a bx c where the coefficient a is a number other than 1 by combining two factoring methods into the same problem. 21

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25 Challenge Problem: Recall that the volume of a rectangular solid (a box) is given by V L W H. If a particular rectangular solid has a volume of 5 15x 10, how would you represent the length, width and height of the solid? Justify your answer. SUMMARY Exit Ticket 23

26 Day 4 Factoring Trinomials Completely Homework 24

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28 Day 5: More Factoring Completely Warm - Up Some polynomials cannot be factored into the product of two binomials with integer coefficients, (such as x ), and are referred to as prime. Other polynomials contain a multitude of factors. "Factoring completely" means to continue factoring until no further factors can be found. More specifically, it means to continue factoring until all factors other than monomial factors are prime factors. You will have to look at the problems very carefully to be sure that you have found all of the possible factors. To factor completely: 1. Search for a greatest common factor. If you find one, factor it out of the polynomial. 2. Examine what remains, looking for a trinomial or a binomial which can be factored. 3. Express the answer as the product of all of the factors you have found. 26

29 Example 1: Factoring Completely FACTOR: 10x 2-40 Practice: Factoring Completely 27

30 Example 2: Factoring Completely Factor: 8 Factor: 2 Practice: Factoring Completely x x x 3-8x x 28

31 Challenge Problem: Summary: Exit Ticket: 29

32 Day 5: Homework 30

33 REVIEW FOR TEST SWBAT: Apply their knowledge on Factoring Factor. Station # 1 Common Monomial Factors (GCF) 1) 9x 2 21x 5 2) 4x 3 6x x 3) Factor by Grouping. 4) 5) Factor. Station # 2 Difference of Two Squares D.O.T.S 1) x² ) 36x² - y² 3) 64 - y² 4) 9a² - 121y² 5) a 6 9b 12 6) 25x 4-144y² 31

34 Station # 3 Factoring Trinomials Diamond 1) x² + 21x ) x² - 10x ) x² + 3x 18 4) x² - 7x ) x² - 6x ) x² - x 56 Station # 4 Factoring Completely 1) ax² - a 2) 4a ) 12x 2 3y² 4) 9a 4 36b 4 5) 3x x 42 6) x 4 3x 3 40x² 32

35 Station # 5 Word Problems 1) The area of rectangle is represented by x 2 + 9x Find the binomials that could represent the lengths and width of the rectangle. 2) The Volume of rectangular prism is represented by p 3-12p p. Find the factors that would represent the length, width, and height of the rectangular prism. 33

36 Chapter 8 Review SWBAT: Apply Their Knowledge on Factoring. A) 24t A) 2y 4 A) 5n 9 B) 3t 6 B) 2y 2 B) 3n 4 C) t 2 C) y 3 C) 15n D) t 6 D) 2y D) 3n 9 4. Factor each expression using the GCF

37 A) (x + 6)(x + 1) A) (x - 3)(x - 7) A) (x + 5)(x + 10) B) (x + 5)(x + 1) B) (x - 3)(x + 7) B) (x 5)(x 3) C) (x - 5)(x + 1) C) (x + 10)(x + 11) C) (x + 5)(x + 3) D) (x + 2)(x + 3) D) (x + 3)(x - 7) D) (x 5)( x + 3) Factor each binomial x A) (b - 8)(b - 2) A) (5x + 2)(5x - 2) B) (b + 4)(b + 4) B) (15x + 2)(10x - 2) C) (b + 8)(b + 2) C) (x + 2)(5x - 2) D) (b - 4)(b + 4) D) (5x + 2)(5x + 2) A) 3x 3 (x 2-9) B) 3x 3 (x + 3)(x - 3) C) 3x 3 (x + 3)(x + 3) D) 9x 3 (x 2-9)

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39 A box has a volume given by the trinomial + 3. What are the possible dimensions of the box? Use factoring completely. a. b. c. d. 37