1. Welcome to MM218! Kirsten Meymaris, Mar 15 th Unit 3 : Factoring Part 2 Plan for the hour Review of Factoring from Unit 2 MML questions from Unit 2 Test for Factorability Sum and Difference of Two Cubes Factoring Strategy Review Solving Quadratic Equations by Factoring

2. Welcome to MM218! Kirsten Meymaris, Mar 7 th Unit 2 : Factoring Part 1 Plan for the hour Factors Factoring by Grouping Factoring Trinomials, x 2 + bx + c Product - Sum Factoring Trinomials, ax 2 + bx + c Product – sum Trail and Error

3. Example

4. Factor by Grouping Look for common factors 3x 2 + 3x + 5x + 5 Factor each group separately 3x (x + 1) + 5 (x + 1) (3x + 5) (x + 1) 3x 2 & 3x have common factors 5x & 5 have common factors (x + 1) are common factors

5. 1)Write the polynomial so that it is in the form ax 2 + bx + c 2) Factor out the GCF if one exists. 3) Using the given polynomial for values of a, b, and c, multiply 'a' times 'c' to get the grouping number 'ac'. 4) Determine the particular factors of 'ac' that will produce 'b' when combined 5) Write bx as the sum of two terms 6) Factor by grouping PRODUCT – SUM Method with Grouping

6. PRODUCT – SUM Method 3x 2 + 8x + 5 Ax 2 + Bx + C PRODUCT is A * C = 3 * 5 = 15 SUM is simply B = 8 Find two numbers whose PRODUCT is 15 and SUM is 8 1*15 or 3*5 3 + 5

7. Substitute the PRODUCT factors for the SUM 3x 2 + 8x + 5 3x 2 + 3x + 5x + 5 Now factor by grouping 3x(x + 1) + 5(x + 1) (3x + 5) (x + 1) PRODUCT – SUM Method

8. x 2 +5x+6 Step 1: First, look at the sign on the last term of the trinomial, which is positive. x 2 +5x+6 Step 2: Since the sign on the last term is positive, we know both signs in the two factors are plus signs or both signs in the two factors are negative signs. Is it ( + )( + ) or ( − )( − ) ? The sign of the middle term is +5x, so we know the right choice is ( + )( + ).

9. Step 3: After determining the signs in the two factors, we turn our attention to what goes on either side of each sign. (x+ )(x+ ) Step 4: Next, we factor the last term of the trinomial and place those factors in the last positions. (x+2)(x+3) Step 5: Finally, FOIL your answer to check to see if you get the original trinomial. x 2 +5x+6 = (x+2)(x+3)

10. x 2 + 6x − 7 Step 1: x 2 + 6x −7 Step 2: Sign on last term is negative, so we know one of the factors has a positive sign and the other factor has a negative sign. Step 3: Factor first term of the trinomial and place those factors in the first position. Step 4: Factor last term of the trinomial and place those factors in the last position. Step 5: FOIL it -- Does it check?

11. Step 1: x 2 +6x−7 Step 2: ( + )( − ) Step 3: (x+ )(x− ) Step 4: (x+7)(x−1) Step 5: FOIL to check: x 2 +6x−7= (x+7)(x−1)

12. 2 Terms Polynomials After you factor out the GCF, polynomials with 2 terms factor one of three ways – Difference of squares – Difference of cubes – Sum of cubes

13. Difference of Squares Meaning: A perfect square minus a perfect square Form: F 2 – L 2 Factors as: (F + L)(F – L) Examples: x 2 – 9 = y 2 – 36 =

14. Difference of Squares Meaning: A perfect square minus a perfect square Form: F 2 – L 2 Factors as: (F + L)(F – L) Examples: x 2 – 9 = x 2 – 3 2 =(x + 3)(x – 3) y 2 – 36 = y 2 – 6 2 =(y + 6)(y – 6)

15. Difference of Cubes Meaning: A perfect cube minus a perfect cube Form: F 3 – L 3 Factors as: (F - L)(F 2 + FL + L 2 ) Examples: x 3 – 8 = x 3 – 64 =

16. Difference of Cubes Meaning: A perfect cube minus a perfect cube Form: F 3 – L 3 Factors as: (F - L)(F 2 + FL + L 2 ) Examples: x 3 – 8 = (x 3 – 2 3 ) = (x – 2)( x 2 - 2x + 4) x 3 – 27 = (x 3 – 3 3 )= (x – 3) (x 2 – 3x + 9)

17. Sum of Cubes Meaning: A perfect cube plus a perfect cube Form: F 3 + L 3 Factors as: (F + L)(F 2 - FL + L 2 ) Examples: x 3 + 8 = x 3 + 27 =

18. Sum of Cubes Meaning: A perfect cube plus a perfect cube Form: F 3 + L 3 Factors as: (F + L)(F 2 - FL + L 2 ) Examples: x 3 + 8 = (x 3 + 2 3 ) = (x + 2)(x 2 – 2x + 4) x 3 + 27 = (x 3 + 3 3 )= (x + 3)(x 2 -3x + 9)

19. Sum of Squares Meaning: A perfect square plus a perfect square Form: F 2 + L 2 Factors as: DOES NOT FACTOR … do not be tempted! Examples: x 2 + 4 x 2 + 100

20. The KEY to these are knowing what perfect squares and cubes are PERFECT SQUARESPERFECT CUBES 1 36 4 49 9 64 16 81 25 100 etc 1 8 27 64 125 etc

21. Examples x 2 – 36 25x 2 – 81y 2 27x 2 - 75

22. Review of Factoring Strategies Common Factor Difference of Two Squares Perfect-square Trinomial Trinomial x 2 + bx +c Trinomial ax 2 + bx +c Factor by grouping

23. Does every polynomial factor? No … some polynomials (or numbers for that matter) do not factor! Things that do not factor are called PRIME

24. Test for Factorability 4x 2 - 9x - 9. Trinomial must be of the form ax 2 + bx + c a = 4 b = - 9 c = -9 plug them into b 2 - 4ac b 2 - 4ac = (-9) 2 - 4(4) (-9) = 81 + (-16) (-9) = 81 + 144 = 225 225 is a perfect square, because 15 15 = 225. Trinomial is factorable! 4x 2 - 9x - 9

25. 1. Write the trinomial in descending powers of one variable. 2. Factor out any greatest common factor (including 1, if that is necessary to make the coefficient of the first term positive. 3. Test the trinomial for factorability. 4. When the sign of the third term is +, the signs between the terms of each binomial factor are the same as the sign of the middle term of the trinomial. When the sign of the third term is -, the signs between the terms of the binomials must be opposite. 5. Try various combinations of the factors of the first terms and the last terms until you find the one that works. 6. Check the factorization by multiplication.

26. Solving Quadratic Equations by Factoring Standard form of quadratic equation: ax 2 + bx + c = 0 (where a and c are not equal to 0) Zero – factor property a * b = 0 if a= 0 or b = 0

27. Examples x 2 – x – 20 = 09x 2 = 81

28. Examples 6x 2 = 16x – 85x 2 + 3x = 8x

29. Thank You! Next up: Unit 4 Introduction to Factoring Remember to Ask, Ask, Ask! kmeymaris@kaplan.edu AIM: kkmeymaris kmeymaris@kaplan.edu