2. A “Difference of Squares” is a binomial ( *2 terms only*) and it factors like this:

3. Factoring a polynomial means expressing it as a product of other polynomials.

4. Factoring polynomials with a common monomial factor (using GCF). **Always look for a GCF before using any other factoring method. Factoring Method #1

5. Steps: 1. Find the greatest common factor (GCF). 2. Divide the polynomial by the GCF. The quotient is the other factor. 3. Express the polynomial as the product of the quotient and the GCF.

6. Step 1: Step 2: Divide by GCF

7. The answer should look like this:

8. Factor these on your own looking for a GCF.

11. Factoring polynomials that are a difference of squares. Factoring Method #2

12. To factor, express each term as a square of a monomial then apply the rule...

13. Here is another example:

16. Try these on your own:

17. Sum and Difference of Cubes:

18. Factoring a Sum of Two Cubes The sum of two cubes has to be exactly in this form to use this rule. 1. When you have the sum of two cubes, you have a product of a binomial and a trinomial. 2.The binomial is the sum of the bases that are being cubed. 3. The trinomial is the first base squared, the second term is the opposite of the product of the two bases found, and the third term is the second base squared.

19. Factor the sum of cubes: Example 1 First note that there is no GCF to factor out of this polynomial. This fits the form of the sum of cubes. The cube root of The cube root of 27 = 3

20. Factor the sum of cubes: (GCF first, if needed) Example 2 5x 3 + 4027x 3 + 8

21. Rewrite as cubes Write each monomial as a cube and apply either of the rules. Apply the rule for sum of cubes:

23. Factoring a Difference of Two Cubes 1. When you have the difference of two cubes, you have a product of a binomial and a trinomial. 2. The binomial is the difference of the bases that are being cubed. 3. The trinomial is the first base squared, the second term is the product of the two bases found, and the third term is the second base squared.

24. Example 4 Factor 8x 3 – y 3 54x 3 – 81y 3

25. Factor the difference of cubes: Example 3 First note that there is no GCF to factor out of this polynomial. This fits the form of the difference of cubes. The cube root of The cube root of 8 = 2

26. Rewrite as cubes Apply the rule for difference of cubes:

29. Factoring Technique #3 Factoring By Grouping for polynomials with 4 or more terms

31. Factoring By Grouping 1. Group the first set of terms and last set of terms with parentheses. 2. Factor out the GCF from each group so that both sets of parentheses contain the same factors. 3. Factor out the GCF again (the GCF is the factor from step 2).

32. Step 1: Group ( + between the parentheses) Example 1: Step 2: Factor out GCF from each group Step 3: Factor out GCF again

33. Example 2:

37. Factoring Method #4 Factoring a trinomial in the form:

41. Next Factoring a trinomial: 2. Product of first terms of both binomials must equal first term of the trinomial. 1. Write two sets of parenthesis, ( )( ). These will be the factors of the trinomial.

42. 3. The product of last terms of both binomials must equal last term of the trinomial (c). 4. Think of the FOIL method of multiplying binomials, the sum of the outer and the inner products must equal the middle term (bx). Factoring a trinomial:

43. x x Factors of +8: 1 & 8 2 & 4 -1 & -8 -2 & -4 2x + 4x = 6x 1x + 8x = 9x O + I = bx ? -2x - 4x = -6x -1x - 8x = -9x -2 -4

44. Check your answer by using FOIL FOIL

45. Lets do another example: Find a GCF Factor trinomial Don’t Forget Method #1. Always check for GCF before you do anything else.

46. Factoring Trinomials The hard case – “Box Method” Note: The c oefficient of x 2 is different from 1. In this case it is 2 First: Multiply 2 and –6: 2 (– 6) = – 12 1 Next: Find factors of – 12 that add up to 1 – 3 and 4

47. Factoring Trinomials The hard case – “Box Method” 1.Draw a 2 by 2 grid. 2.Write the first term in the upper left-hand corner 3.Write the last term in the lower right-hand corner.

48. – 3 x 4 = – 12 – 3 + 4 = 1 1.Take the two numbers –3 and 4, and put them, complete with signs and variables, in the diagonal corners, like this: It does not matter which way you do the diagonal entries! Find factors of – 12 that add up to 1 –3 x 4 x

49. The hard case – “Box Method” 1.Then factor like this: Factor Top RowFactor Bottom Row From Left ColumnFrom Right Column

50. The hard case – “Box Method” Note: The signs for the bottom row entry and the right column entry come from the closest term that you are factoring from. DO NOT FORGET THE SIGNS!! + + Now that we have factored our box we can read off our answer:

51. The hard case – “Box Method” Finally, you can check your work by multiplying back to get the original answer. Look for factors of 48 that add up to –19 – 16 and – 3

52. Factoring Technique #3 Factoring a perfect square trinomial in the form:

53. Perfect Square Trinomials can be factored just like other trinomials (guess and check), but if you recognize the perfect squares pattern, follow the formula!

54. Does the middle term fit the pattern, 2ab? Yes, the factors are (a + b) 2 : b a

55. Does the middle term fit the pattern, 2ab? Yes, the factors are (a - b) 2 : b a

56. Try these on your own:

57. Answers: