1. Martin-Gay, Developmental Mathematics 1 Name: Date: Topic: Factoring Polynomials (Special Cases & By Grouping) Essential Question: How can you factor special case trinomials and how does it compare to multiplying special case binomials? Warm-Up: Factor each expression  6x 2 + 13x + 5  10x 2 + 31x – 14

2. Martin-Gay, Developmental Mathematics 2 Home-Learning Review

3. Martin-Gay, Developmental Mathematics 3 Quiz #11

4. Martin-Gay, Developmental Mathematics 4 Factoring Polynomials Special Cases & By Grouping

5. Martin-Gay, Developmental Mathematics 5 Difference of Squares When factoring using a difference of squares, look for the following three things:  only 2 terms  minus sign between them  both terms must be perfect squares If all 3 of the above are true, write two ( ), one with a + sign and one with a – sign: ( + ) ( - ). Examples: z 2 – 9 16x 2 – 81 24g 2 – 6

6. Martin-Gay, Developmental Mathematics 6 A “Difference of Squares” is a binomial ( *2 terms only*) and it factors like this: Difference of Squares

7. Martin-Gay, Developmental Mathematics 7 To factor, express each term as a square of a monomial then apply the rule... Difference of Squares

8. Martin-Gay, Developmental Mathematics 8 Examples:  z 2 – 9  16x 2 – 81  24g 2 – 6 Lets Try Together

9. Martin-Gay, Developmental Mathematics 9 Try These 1. a 2 – 16 2. x 2 – 25 3. 4y 2 – 16 4. 9y 2 – 25

10. Martin-Gay, Developmental Mathematics 10 Factoring Four Term Polynomials by Grouping

11. Martin-Gay, Developmental Mathematics 11 Factor by Grouping  When polynomials contain four terms, it is sometimes easier to group like terms in order to factor.  Your goal is to create a common factor.  You can also move terms around in the polynomial to create a common factor.  Practice makes you better in recognizing common factors.

12. Martin-Gay, Developmental Mathematics 12 Factor by Grouping Example 1:  FACTOR: 3xy - 21y + 5x – 35  Factor the first two terms: 3xy - 21y = 3y (x – 7)  Factor the last two terms: + 5x - 35 = 5 (x – 7)  The green parentheses are the same so it’s the common factor Now you have a common factor (x - 7) (3y + 5)

13. Martin-Gay, Developmental Mathematics 13 Factor by Grouping Example 2:  FACTOR: 6mx – 4m + 3rx – 2r  Factor the first two terms: 6mx – 4m = 2m (3x - 2)  Factor the last two terms: + 3rx – 2r = r (3x - 2)  The green parentheses are the same so it’s the common factor Now you have a common factor (3x - 2) (2m + r)

14. Martin-Gay, Developmental Mathematics 14 Factor by Grouping Example 3:  FACTOR: 15x – 3xy + 4y –20  Factor the first two terms: 15x – 3xy = 3x (5 – y)  Factor the last two terms: + 4y –20 = 4 (y – 5)  The green parentheses are opposites so change the sign on the 4 - 4 (-y + 5) or – 4 (5 - y)  Now you have a common factor (5 – y) (3x – 4)

15. Martin-Gay, Developmental Mathematics 15 Factoring Completely Now that we’ve learned all the types of factoring, we need to remember to use them all. Whenever it says to factor, you must break down the expression into the smallest possible factors. Let’s review all the ways to factor.

16. Martin-Gay, Developmental Mathematics 16 Types of Factoring 1.Look for GCF first. 2.Count the number of terms: a) 4 terms – factor by grouping b) 3 terms – apply strategies of factorization >x 2 + bx + c or ax 2 + bx + c c) 2 terms - look for difference of squares

17. Martin-Gay, Developmental Mathematics 17 Solving Equations by Factoring 1.We know that an equation must be solved for the unknown. 2.Up to now, we have only solved equations with a degree of 1. 2x + 8 = 4x +6 -2x + 8 = 6 -2x = -2 x = 1

18. Martin-Gay, Developmental Mathematics 18 Steps to Solve Equations by Factoring 3.If an equation has a degree of 2 or higher, we cannot solve it until it has been factored. 4.You must first get “0” on one side of the = sign before you try any factoring. 5.Once you have “0” on one side, use all your rules for factoring to make 2 ( ) or factors.

19. Martin-Gay, Developmental Mathematics 19 Steps to Solve Equations by Factoring 6.Next, set each factor = 0 and solve for the unknown. x 2 + 12x = 0 Factor GCF x(x + 12)(x – 3) = 0 (set each factor = 0, & solve) x = 0 x + 12 = 0 x – 3 = 0 x = - 12 x = 3 7.You now have 3 answers, x = 0, x = -12, and x = 3.

20. Martin-Gay, Developmental Mathematics 20 Time to Practice: Page 515 – 516 (24, 31, 40) Page 519 – 521 (1, 2, 47 – 50) Solve for x: x 2 – 10x – 24 2x 2 + 13x + 6