1. Factoring – Trinomials (a ≠ 1), Guess and Check It is assumed you already know how to factor trinomials where a = 1, that is, trinomials of the form Be sure to study the previous slideshow if you are not confident in factoring these trinomials.

2. The process is very similar to the a = 1 pattern with a little bit more work. The method discussed in this slideshow could be called “Guess and Check.” We now turn our attention to factoring trinomials of the form We consider the various options for coefficients and check each one until the solution is found.

3. Another procedure for factoring these more difficult trinomials is called the “ac method.” That method is discussed in another slide show. You only need to know one of these methods, though it can be handy to know both. While at times the guess and check method can be faster, the ac method is very straightforward without all the guessing. It is suggested that you look at both and determine which is easiest for you.

4. Guess and Check Method To factor a trinomial of the form 3.Determine the possible factors of c. These will be the last terms. 2.Determine the signs 4.Try the various combinations until the outside/inside term from the binomials is bx 1.Determine the possible factors of a. These will be the first terms.

5. Example 1 Factor: 1.Determine the possible factors of a. These will be the first terms. 2.Determine the signs

6. 3.Determine the possible factors of c. These will be the last terms. 4.Try the various combinations until the outside/inside term from the binomials is bx

7. No Outside/Inside

8. Now comes the major difference in the a ≠ 1 pattern. Switch around the 1 and the 3, and check the outside/inside again. No Yes

9. The trinomial is factored using

10. Same numerical value, possibly opposite in sign. Notice a very important difference in the a = 1 and the a ≠ 1 cases. Possible FactorsSwitch Last terms Outside/Inside

11. Different numerical values! Possible FactorsSwitch Last terms Outside/Inside

12. In the a = 1 case In the a ≠ 1 case switching the last terms of the binomials will not change the numerical value of the outside/inside term. In some instances it may change the sign. switching the last terms of the binomials will usually change the numerical value of the outside/inside term, and possibly the sign. In the a ≠ 1 case it is important to switch the last terms to check all possibilities.

13. Example 2 Factor: 1.Determine the possible factors of a. These will be the first terms. 2.Determine the signs

14. 3.Determine the possible factors of c. These will be the last terms. 4.Try the various combinations until the outside/inside term from the binomials is bx

15. No Last Terms Factors Outside/ Inside Middle Term

16. None of the combinations worked to give us the correct middle term. Try the other pair of numbers for the first term Recall that there were two possible combinations for the first term. and repeat the process with the last terms.

17. No Yes Last Terms Factors Outside/ Inside Middle Term

18. The trinomial is factored using All of this may seem rather long and difficult, but many of the steps can be completed in your head, as will be seen in the next example.

19. Example 3 Possible first factors Possible last factors Hint: start with the bottom pair in each list and work your way up.

20. First Signs No Check Last

21. Right number, wrong sign Check Switch Last Switch signs

22. The trinomial is factored using Notice that this time we got “lucky” and found the answer rather quickly. There were a number of combinations to try, and we found the correct answer on the second try.

23. Switch Last Here is a good way to quickly determine all possible combinations: Factors of aFactors of c Each first pair matched up with each last pair

24. Here is a good way to quickly determine all possible combinations: Each first pair matched up with each last pair

25. Here is a good way to quickly determine all possible combinations: Each first pair matched up with each last pair

26. This amounted to 12 different combinations! While it can be a lot of work to check the outside/inside on each combination, most of them can be eliminated very quickly. For example: This combination isn’t even close, and can be eliminated without doing any of the math.