Factoring Trinomials a 1 Guess and Check It

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. Table of Contents

• 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. Table of Contents

• 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. Table of Contents

Guess and Check Method To factor a trinomial of the form 1. Determine the possible factors of a. These will be the first terms. 2. Determine the signs 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 Table of Contents

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

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 Table of Contents

Outside/Inside No Table of Contents

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 Table of Contents

The trinomial is factored using Table of Contents

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

Possible Factors Switch Last terms Outside/Inside Different numerical values! Table of Contents

• 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. Table of Contents

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

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 Table of Contents

Last Terms Factors Outside/ Inside Middle Term No No Table of Contents

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. Table of Contents

Last Terms Factors Outside/ Inside Middle Term No No No Yes Table of Contents

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. Table of Contents

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

Signs First Check Last No Table of Contents

Switch Last Check Right number, wrong sign Switch signs Table of Contents

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. Table of Contents

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

• Here is a good way to quickly determine all possible combinations: Each first pair matched up with each last pair Table of Contents

• Here is a good way to quickly determine all possible combinations: Each first pair matched up with each last pair Table of Contents

• 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. Table of Contents

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