1. 5-4 Factoring Polynomials
Objectives:
Students will be able to:
Factor polynomials
Simplify polynomial quotients by factoring

2. Factoring
There are various different techniques used to factor polynomials.
The technique(s) used depend on the number of terms in the polynomial, and what those terms are.
Throughout this section we will examine different factoring techniques and how to utilize one or more of those techniques to factor a polynomial.

3. What is a GCF
Greatest common factor (GCF): largest factor that all terms have in common
You can find the GCF for a polynomial of two or more terms.

4. Example 1: Finding a GCF
Example 1: Find the GCF of each set of monomials.
8, 12 b) 10, 21 c) 24, 60, 36 4 1 12

5. Ex 1: Finding GCFs
2x2
6a2b
3xy2 4x
3x2

6. Factoring a Polynomial w/GCF
Determine what the GCF of the terms is, and factor that out
Rewrite the expression using the distributive property

7. Ex 2: Factoring By Distributive Property
Factor each polynomial.

8. Try these.

9. Grouping
Grouping is a factoring technique used when a polynomial contains four or more terms.

10. Steps for Factoring By Grouping
Group terms with common factors (separate the polynomial expression into the sum of two separate expressions)
Factor the GCF out of each expression
Rewrite the expression using the distributive property (factor into a binomial multiplied by a binomial)

11. Example 3: Factor each polynomial.

12. Ex 3: Factor each polynomial.

13. Ex 3: Continued.

14. Ex 3: Cont.

15. Factoring Trinomials The standard form for a trinomial is:
The goal of factoring a trinomial is to factor it into two binomials. [If we re-multiplied the binomials together, that should get us back to the original trinomial.]

16. Steps to factor a Trinomial
Steps for factoring a trinomial
Multiply a * c
2) Look for factors of the product in step 1 that add to give you the ‘b’ term.
3) Rewrite the ‘b’ term using these two factors.
4) Factor by grouping.

17. Ex4: Factoring Trinomials

18. Example 4: Factor each polynomial

19. Try some more…

20. Try some more…

21. Try these.

22. More Examples

23. More Examples

24. Look For GCF first!
There are instances when a polynomial will have a GCF that can be factored out first. Doing so will make factoring a trinomial much easier.

25. Ex 5: Factor each polynomial

26. Ex 5: Factor each polynomial

27. Ex 5: GCF first!

28. Additional Factoring Techniques
There are certain binomials that are factorable, but cannot be factored using any of the previous factoring techniques.
These binomials deal with perfect square factors or perfect cube factors.

29. Factoring Differences of Squares

30. Factoring Differences of Squares

31. Factoring Differences of Squares

32. Factoring Differences of Squares
GCF first!!

33. Factoring Differences of Squares

34. Factoring Differences of Squares

35. Sum/Difference of Cubes

36. Sum/Difference of Cubes

37. Try these

38. Try these

39. Try these

40. Simplifying Polynomial Quotients
In the previous section (5-3), we learned to simplify the quotient of two polynomials using long division or synthetic division. Some quotients can be simplified using factoring. To do so: 1) factor the numerator (if possible) 2) factor the denominator (if possible) 3) reduce the fraction TIP: Be sure to check for values that the variable cannot equal. Remember that the denominator of a fraction can never be zero.

41. Ex1: Simplify Factor Numerator and Denominator!
Eliminate Common Factors in Numerator and Denominator!

42. Ex 2: Simplify

43. Ex 3: Simplify
In order to eliminate common factors , one must be in the numerator an the other in the denominator. This expression cannot be simplified further…

44. To recap:
Always try and factor out a GCF first, if possible. It will make life much easier.