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Factoring – Trinomials ( a ≠ 1), Guess and Check

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.

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Factoring – Trinomials ( a ≠ 1), Guess and Check

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  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. We now turn our attention to factoring trinomials of the form • 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 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 • Determine the possible factorsof a. These will be the first terms. • Determine the signs • Determine the possible factorsof c. These will be the last terms. • Try the various combinations until the outside/inside term from the binomials is bx

  5. Example 1 Factor: • Determine the possible factorsof a. These will be the first terms. • Determine the signs

  6. Determine the possible factorsof c. These will be the last terms. • Try the various combinations until the outside/inside term from the binomials is bx

  7. Outside/Inside No

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

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

  12. 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. • In the a ≠ 1 case 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: • Determine the possible factorsof a. These will be the first terms. • Determine the signs

  14. Determine the possible factorsof c. These will be the last terms. • Try the various combinations until the outside/inside term from the binomials is bx

  15. Last Terms Outside/Inside Middle Term Factors No No No No

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

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

  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. Last First Signs Check No

  21. Switch Last Check Right number, wrong sign 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. Here is a good way to quickly determine allpossible combinations: Factors of a Factors of c Switch Last Each first pairmatched up witheach last pair

  24. Here is a good way to quickly determine allpossible combinations: Each first pairmatched up witheach last pair

  25. Here is a good way to quickly determine allpossible combinations: Each first pairmatched up witheach 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.

  27. END OF PRESENTATION

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