The document discusses various methods for factoring polynomials, including factoring by grouping terms, factoring trinomials, factoring the difference of two squares, factoring perfect square trinomials, factoring the sum and difference of two cubes, and factoring expressions with fractional and negative exponents. Examples are provided to illustrate each factoring technique.

1. Section P5 Factoring Polynomials

2. Common Factors

3. Factoring a polynomial containing the sum of monomials mean finding an equivalent expression that is a product. In this section we will be factoring over the set of integers, meaning that the coefficients in the factors are integers. Polynomials that cannot be factored using integer coefficients are called prime.

4. Example Factor:

5. Factoring by Grouping

6. Sometimes all of the terms of a polynomial may not contain a common factor. However, by a suitable grouping of terms it may be possible to factor. This is called factoring by grouping.

7. Example Factor by Grouping:

8. Example Factor by Grouping:

9. Factoring Trinomials

11. Factor: 4 2 + + Choose either two positive or two negative factors since the sign in front of the 8 is positive. -6 -9 6 9 Sum of Factors -4,-2 -8,-1 4,2 8,1 Factors of 8

12. Factor: + - 1 5 Since the sign in front of the 5 is a negative, one factor will be positive and one will be negative. Sum of outside and inside products Possible factorizations

13. Example Factor: Sum of Inside and Outside products Possible Factorizations

14. Example Factor: Sum of Inside and outside Products Possible Factorizations

15. Factoring the Difference of Two Squares

18. Repeated Factorization- Another example Can the sum of two squares be factored?

19. Example Factor Completely:

20. Example Factor Completely:

21. Example Factor Completely:

22. Factoring Perfect Square Trinomials

24. Example Factor:

25. Example Factor:

26. Factoring the Sum and Difference of Two Cubes

28. Example Factor:

29. Example Factor:

30. Example Factor:

31. A Strategy for Factoring Polynomials

33. Example Factor Completely:

34. Example Factor Completely:

35. Example Factor Completely:

36. Example Factor Completely:

37. Example Factor Completely:

38. Example Factor Completely:

39. Factoring Algebraic Expressions Containing Fractional and Negative Exponents

40. Expressions with fractional and negative exponents are not polynomials, but they can be factored using similar techniques. Find the greatest common factor with the smallest exponent in the terms.

41. Example Factor and simplify:

42. Example Factor and simplify:

43. Example Factor and simplify:

44. (a) (b) (c) (d) Factor Completely:

45. (a) (b) (c) (d) Factor Completely: