table of contents factoring – trinomials (a ≠ 1), guess and check it is assumed you already know...
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Table of Contents
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.
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• 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.
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• 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.
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Guess and Check Method
To factor a trinomial of the form2ax 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.
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• 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
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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
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22 7 3x x
( 1)(2 3)x x
3 2 5x x x
No
Outside/Inside
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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
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The trinomial is factored using
22 7 3x x
( 3)(2 1)x x
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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
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23 10x x
Different numerical values!
1a
3 2 5x x
Possible Factors Switch Last terms
13xOutside/Inside
1xOutside/Inside
3 5 2x x
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• 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.
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• Example 2
1 10
2 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
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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 6
2 3
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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
FactorsOutside/Inside
MiddleTerm
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None of the combinations worked to give us the correct middle term.
Try the other pair of numbers for the first term
1 10
2 5
Recall that there were two possible combinations for the first term.
(2 )(5 )x x
and repeat the process with the last terms.
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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
FactorsOutside/Inside
MiddleTerm
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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.
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• Example 3
1 12
2 6
3 4
212 13 35x x
1 35
5 7
Possible first factors
Possible last factors
Hint: start with the bottom pair in each list and work your way up.
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1 12
2 6
3 4
212 13 35x x 1 35
5 7
(3 )(4 )x x
FirstSigns
NoCheck
(3 )(4 )x x Last
(3 5)(4 7)x x
21 20x x x
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1 12
2 6
3 4
212 13 35x x 1 35
5 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
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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.
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1 12
2 6
3 4
1 35
5 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 1
7 5
Each first pair matched up with each last pair
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1 12
2 6
3 4
1 35
5 7
• Here is a good way to quickly determine all possible combinations:
212 13 35x x
35 1
7 5
Each first pair matched up with each last pair
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1 12
2 6
3 4
1 35
5 7
• Here is a good way to quickly determine all possible combinations:
212 13 35x x
35 1
7 5
Each first pair matched up with each last pair
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• 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.
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