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

2x bx c • Be sure to study the previous slideshow if you are not

confident in factoring these trinomials.

• 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

2 , 1ax bx c a

• We consider the various options for coefficients and check each one until the solution is found.

• 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.

Guess and Check Method

To factor a trinomial of the form 2ax bx c

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.

• Example 1

1 2

22 7 3x x Factor:

1 2x x

1. Determine the possible factors of a. These will be the first terms.

2. Determine the signs 2x x

1 3

22 7 3x x

2x x

3. Determine the possible factors of c. These will be the last terms. 1 2 3x x

4. Try the various combinations until the outside/inside term from the binomials is bx

22 7 3x x

( 1)(2 3)x x

3 2 5x x x

No

Outside/Inside

Now comes the major difference in the a ≠ 1 pattern. Switch around the 1 and the 3, and check the outside/inside again.

22 7 3x x

( 1)(2 3)x x

3 2 5x x x

No

( 3)(2 1)x x

6 7x x x

Yes

The trinomial is factored using

22 7 3x x

( 3)(2 1)x x

2 4 12x x

Same numerical value, possibly opposite in sign.

• Notice a very important difference in the a = 1 and the a ≠ 1 cases.

1a

2 6x x 6 2x x

Possible Factors Switch Last terms

4xOutside/Inside

4xOutside/Inside

23 10x x

Different numerical values!

1a

3 2 5x x

Possible Factors Switch Last terms

13xOutside/Inside

1xOutside/Inside

3 5 2x x

• 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.

• Example 2

1 102 5

Factor:

1 10x x

1. Determine the possible factors of a. These will be the first terms.

2. Determine the signs

210 19 6x x

10x x

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

210 19 6x x

1 62 3

210 19 6x x

( 1)(10 6)x x 16x No

( 6)(10 1)x x 61x

1,6

6,1 No

( 2)(10 3)x x 23x No

( 3)(10 2)x x 32x

2,3

3,2 No

Last Terms Factors Outside/

InsideMiddleTerm

None of the combinations worked to give us the correct middle term.

Try the other pair of numbers for the first term

1 102 5

Recall that there were two possible combinations for the first term.

(2 )(5 )x x

and repeat the process with the last terms.

210 19 6x x

(2 1)(5 6)x x 17x No

(2 6)(5 1)x x 32x

1,6

6,1 No

(2 2)(5 3)x x 16x No

(2 3)(5 2)x x 19x

2,3

3,2 Yes

Last Terms Factors Outside/

InsideMiddleTerm

The trinomial is factored using

210 19 6x x

(2 3)(5 2)x x

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.

• Example 3

1 122 63 4

212 13 35x x

1 355 7

Possible first factors

Possible last factors

Hint: start with the bottom pair in each list and work your way up.

1 122 63 4

212 13 35x x 1 355 7

(3 )(4 )x x

FirstSigns

NoCheck

(3 )(4 )x x Last

(3 5)(4 7)x x

21 20x x x

1 122 63 4

212 13 35x x 1 355 7

Right number, wrong sign

Check

(3 5)(4 7)x x

15 28 13x x x

(3 7)(4 5)x x Switch Last

Switch signs (3 7)(4 5)x x

The trinomial is factored using

212 13 35x x

(3 7)(4 5)x x

• 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.

1 122 63 4

1 355 7

Switch Last

• Here is a good way to quickly determine all possible combinations:

212 13 35x x

Factors of a Factors of c

35 17 5

Each first pair matched up with each last pair

1 122 63 4

1 355 7

• Here is a good way to quickly determine all possible combinations:

212 13 35x x

35 17 5

Each first pair matched up with each last pair

1 122 63 4

1 355 7

• Here is a good way to quickly determine all possible combinations:

212 13 35x x

35 17 5

Each first pair matched up with each last pair

• 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:

212 13 35x x

(1 35)(12 1)x x

This combination isn’t even close, and can be eliminated without doing any of the math.