1. Monday, June 30 Factoring

2. Factoring out the GCF

3. Greatest Common Factor The greatest common factor (GCF) is the product of what both items have in common. Example:18xy, 36y 2 18xy = 2 · 3 · 3 · x · y 36y 2 = 2 · 2 · 3 · 3 · y · y GCF = = 18y 2 · 3 · y

4. Now you try! Example 1:12a 2 b, 90a 2 b 2 c Find the greatest common factor of the following: Example 2:15r 2, 35s 2, 70rs GCF = 6a 2 b GCF = 5

5. Factoring - Opposite of distributing - Breaking down a polynomial to what multiplies together to form the polynomial

6. Example: Factor:12a2 + 16a = 2·2·3·a·a + 2·2·2·2·a = 2 · 2 · a (3·a + 2·2) = 4a (3a + 4) You can check by distributing. 1. Factor each term. 2. Pull out the GCF. 3. Multiply.

7. Example: Factor: 18cd 2 + 12c 2 d + 9cd = 2·3·3·c·d·d + 2·2·3·c·c·d + 3·3·c·d = 3 · c · d (2·3·d + 2·2·c + 3) = 3cd (6d + 4c + 3)

8. Now you try! Example 1: 15x + 25x 2 Example 2: 12xy + 24xy 2 – 30x 2 y 4 = 6xy(2 + 4y – 5xy 3 ) = 5x(3 + 5x)

9. Factoring by Grouping

10. Example: Factor:5xy – 35x + 3y – 21 (5xy – 35x) + (3y – 21) = (5·x·y – 5·7·x)+ (3·y – 3·7) = 5·x (y – 7)+ 3 (y – 7) = 5x (y – 7)+ 3 (y – 7) = (5x + 3)(y – 7)

11. Example: Factor:5xy – 35x + 3y – 21 (5xy – 35x) + (3y – 21) = 5x (y – 7)+ 3 (y – 7) = (5x + 3)(y – 7) 1. Group terms with ( ). 2. Pull out GCF from each group. 3. Split into factors.

12. Notes - What is in parentheses MUST be the same!! - Grouping only works if there are 4 terms!!

13. Now you try! Factor. Example 1:5y 2 – 15y + 4y - 12 Example 2:5c – 10c 2 + 2d – 4cd = (5y + 4)(y – 3) = (5c + 2d)(1 – 2c)

14. 2 more important examples: Example 1:2xy + 7x + 2y + 7 (2xy + 7x) + (2y + 7) + (2y + 7) = x (2y + 7) = (x + 1) + 1(2y + 7) (2y + 7)

15. Example 2:15a – 3ab – 20 + 4b (15a – 3ab) – (20 + 4b) – 4 (5 – b)= 3a (5 – b) = (3a – 4)(5 – b) – (15a – 3ab) – (20 – 4b) If there is a negative in the middle, you MUST change the sign after it.

16. Factoring Trinomials

17. Example 1: Factor: x 2 + 5x + 6 6 1 · 6 2 · 3 Look for factors of 6 thatADD topositive 5 (x + 2)(x + 3)

18. Example 2: Factor: x 2 + 7x + 12 12 1 · 12 2 · 6 Look for factors of 12 thatADD topositive 7 (x + 3)(x + 4) 3 · 4

19. Now you try! Example: x 2 + 6x + 8 Example: x 2 + 11x + 10 (x + 2)(x + 4) (x + 1)(x + 10)

20. To determine the signs: Last sign PositiveNegative ( + )( – ) Middle sign PositiveNegative ( + )( + )( – )( – )

21. Example 3: Factor: x 2 – 12x + 27 27 1 · 27 3 · 9 Look for factors of 27 thatADD tonegative 12 (x – 3)(x – 9)

22. Example 4: Factor: x 2 + 3x – 18 18 1 · 18 2 · 9 Look for factors of 18 thatSUBTRACT topositive 3 (x + 6)(x – 3) 3 · 6

23. Now you try! Example: x 2 – x – 20 Example: x 2 – 7x – 18 (x + 4)(x – 5) (x + 2)(x – 9)

24. Please note! Example: x 2 – 5x – 6 Example: x 2 – 5x + 6 (x + 1)(x – 6) (x – 2)(x – 3)

25. More Factoring Trinomials

26. Example 1: Factor: 6x 2 + 17x + 5 30 1 · 30 2 · 15 3 · 10 5 · 6 6x 2 + 2x + 15x + 5 (6x 2 + 2x) + (15x + 5) 2x(3x + 1) + 5(3x + 1) (2x + 5)(3x + 1)

27. Example 2: Factor: 4x 2 + 24x + 32 Always check your factors to see if there is anything more that can be factored out.

28. OR Example 2: Factor: 4x 2 + 24x + 32 It is usually faster if you factor out the GCF first. Always check to see if there is anything you can factor out first.

29. Now you try! Example: 5x 2 + 27x + 10 Example: 24x 2 – 22x + 3 (5x + 2)(x + 5) (4x – 3)(6x – 1)