2. Factoring Expressions This topic will be used throughout the 2 nd semester. Weekly Quizzes

3. What is Factoring? Quick Write: In your notes, write down everything you know about Factoring from Algebra-1 and Geometry. You can use Bullets or give examples 2 Minutes

4. Objectives I can factor expressions using all of these methods –Greatest Common Factor (GCF) –Difference of Two Squares –Reverse FOIL –Swing and Divide I can solve a quadratic equation using factoring

5. Factoring? Factors are two or more terms that multiply to form a product (Factor) x (Factor) = Product Some numbers are Prime, meaning they are only divisible by themselves and 1

6. Method 1 Greatest Common Factor (GCF) – the greatest factor shared by two or more terms ALWAYS try this factoring method 1 st before any other method Divide Out the Biggest common number/variable from each of the terms

7. Greatest Common Factors aka GCF’s Find the GCF for each set of following numbers. Find means tell what the terms have in common. Hint: list the factors and find the greatest match. a)2, 6 b)-25, -40 c)6, 18 d)16, 32 e)3, 8 2 -5 6 16 1 No common factors? GCF =1

8. Find the GCF for each set of following numbers. Hint: list the factors and find the greatest match. a)x, x 2 b)x 2, x 3 c)xy, x 2 y d)2x 3, 8x 2 e)3x 3, 6x 2 f)4x 2, 5y 3 x x2x2 xy 2x 2 Greatest Common Factors aka GCF’s 3x 2 1 No common factors? GCF =1

9. Factor out the GCF for each polynomial: Factor out means you need the GCF times the remaining parts. a)2x + 4y b)5a – 5b c)18x – 6y d)2m + 6mn e)5x 2 y – 10xy 2(x + 2y) 6(3x – y) 5(a – b) 5xy(x - 2) 2m(1 + 3n) Greatest Common Factors aka GCF’s How can you check?

10. Ex 1 15x 2 – 5x GCF = 5x 5x(3x - 1)

11. Ex 2 8x 2 – x GCF = x x(8x - 1)

12. Method #2 Difference of Two Squares a 2 – b 2 = (a + b)(a - b)

13. What is a Perfect Square Any term you can take the square root evenly (No decimal) 25 36 1 x 2 y 4

14. Difference of Perfect Squares x 2 – 4= the answer will look like this: ( )( ) take the square root of each part: ( x 2)(x 2) Make 1 a plus and 1 a minus: (x + 2)(x - 2 )

15. FACTORING Difference of Perfect Squares EX: x 2 – 64 How: Take the square root of each part. One gets a + and one gets a -. Check answer by FOIL. Solution: (x – 8)(x + 8)

16. Example 1 (9x 2 – 16) (3x + 4)(3x – 4)

17. Example 2 x 2 – 16 (x + 4)(x –4)

18. Ex 3 36x 2 – 25 (6x + 5)(6x – 5)

19. Ex 4 9x 2 + 25 PRIME

20. FOIL Review

21. Reverse FOIL When factoring a trinomial with 3 terms you will always get two factors x 2 + bx + c

22. Factoring a Trinomial

24. FOIL Method The two numbers # and * must multiply together to equal 6 The two numbers # and * must add up to 5

25. Example 1 Factoring Tree

26. Example 2 Factoring Tree

27. Example 3 Factoring Tree

28. Swing and Divide Method The Swing & Divide method is very similar to Reverse FOIL, but with 2 extra steps: You can use this method when the number in front of the x 2 term is not 1 EX: 2x 2 – x - 3

29. Swing & Divide 2 Steps: Swing and Divide Don’t do one without doing the 2nd

30. Swing & Divide Method Swing Now use FOIL Divide by Swing # Final Factors

31. Reducing and Checking Always reduce fractions before finding final factors Use TWO Finger Check when done

32. Example 2 Swing Now use FOIL Divide by Swing # Final Factors

33. Example 3 Swing Now use FOIL Divide by Swing # Final Factors Reduce

34. Board 1

35. Board 2

36. Board 3

37. More than ONE Method It is very possible to use more than one factoring method in a problem Remember: ALWAYS use GCF first

38. Example 1 2b 2 x – 50x GCF = 2x 2x(b 2 – 25) 2 nd term is the diff of 2 squares 2x(b + 5)(b - 5)

39. Example 2 32x 3 – 2x GCF = 2x 2x(16x 2 – 1) 2 nd term is the diff of 2 squares 2x(4x + 1)(4x - 1)

40. Equations To solve an equation by factoring, the equation MUST be set equal to ZERO first: ax 2 + bx + c = 0

41. Zero Product Property If ab = 0, then a = 0 b = 0

42. Solving with Factoring Set each variable factor equal to zero and solve for “x”

43. Factoring for solutions x 2 – 3x –4 = 0 (x – 4)(x + 1) = 0 Then by the Zero Product Property: (x – 4) = 0 or (x + 1) = 0 If we solve these for “x” we get the following solutions: x – 4 = 0, so x = 4 x + 1 = 0, so x = -1 These are the two solutions {-1, 4}

44. Example 2

45. Example 3

46. Example 4

47. Homework WS 5-1