Factoring Trinomials by Trial & Error
Factoring Trinomials by Trial & Error
ax2 + bx + c
Consider the product: (6x + 5)(2x + 3) = 12x2 + 18x + 10x + 15 = 12x2 + 28x + 15
[pic] (6x + 5)(2x + 3) = [pic] which is of the form ax2+bx+c
This shows us that polynomials of the form ax2 + bx + c are the result of multiplying using the Foil method.
The two most obvious patterns are:
ax2 represents the First Product
c represents the Last Product
[pic] ax2 + bx + c = (_x + _)(_x+_)
[pic]
Ex: Factor: 2x2 + 11x + 12
2x2 +11x +12 = (_x + _)(_x+_)
[pic]
We need to use factors of 2 as the binomial coefficients and factors of 12 as the binomial constants. These possibilities are given in tabular form:
Factors of (2) Factors of (12)
1,2 3, 4
4, 3
2, 6
6, 2
1, 12
12, 1
Combo #1: (1x+ 3)(2x + 4) = 2x2 + 10x + 12 (no)
Combo #2: (1x+ 4)(2x + 3 ) = 2x2 + 11x+ 12 (yes!)
[pic][pic] 2x2 + 11x + 12 = (x+ 4)(2x + 3 )
There are some strategies that can save you some time in that you may not have to try all combinations.
Ex: Factor 8x2 + 19x + 6
8x2 +19x +6 = (_x + _)(_x+_)
[pic]
Remember that you need to first factor out the gcf, if there is one! This polynomial does not have one.
We can find the following factors:
Factors of (8) Factors of (6)
2, 4 2, 3
3, 2
1, 6
1, 8 6, 1
Combo #1: (2x + 2)(4x + 3) This cannot be correct since 2x +2 has a gcf. Since the polynomial we are trying to factor does not have a gcf it cannot be the correct factorization.
Combo #2: (2x + 3)(4x + 2) Again this cannot be correct since 4x + 2 has a gcf.
Combo #3: (2x + 1)(4x + 6) Cannot be correct as 4x + 6 has a gcf.
Combo #4: (2x + 6)(4x + 1) Again a gcf!
Combo #5: (x + 2)(8x + 3) Check the O + I to get 3x + 16x which is 19x!
[pic][pic] 8x2 + 19x + 6 = (x + 2)(8x + 3)
With practice you will not usually have to write down the entire columns of the possible combinations. But, if you are having difficulty getting the correct factorization, then you may want to write them down.
Ex: Factor 8x2 + 26x + 15
[pic] 8x2 + 26x + 15 = (_x + _)(_x+_)
Factors of 8 Factors of 15
2, 4 3, 5 [pic] (2x + 3)(4x + 5) no
5, 3 [pic]or (2x + 5)(4x + 3) Yes!
1, 8 1, 15
15, 1
Ex: Factor 6x2 - 19x + 8
[pic] 6x2 - 19x + 8 = (_x + _)(_x+_)
[pic]
Factors of 6 Factors of 8
2, 3 2, 4 [pic] (2x - 2)(3x - 4) no, GCF!
4, 2 [pic]or (2x -4)(3x - 2) no, GCF!
1, 6 1, 8 [pic] (2x - 1)(3x - 8) Yes!!
8, 1
Ex: Factor 20x2 - 24x – 9
Note that signs must be opposite!
[pic] 20x2 - 24x - 9 = (_x + _)(_x+_)
[pic]
Factors of 20 Factors of -9 Note: a total of 4 combinations!
4,5 -3,3 (4x – 3)(5x + 3) No
-1, 9 (4x – 1)(5x + 9) No
2, 10 (2x – 3)(10x + 3) Yes!
-----------------------
Factors of a
Called binomial coefficants
Factors of c,
Called binomial constants
This gives us all of the possible ways to get the correct First term (ax2 which is 2x2) and Last term (c which is 12).
Essentially, you try the various combinations until you find the correct combination that yields the middle term in your polynomial which in this case is 11x.
***Note how the second column has the numbers written in both orders – we will see that the different order yields a different middle term. To get all combinations we need only reverse one of the columns.
This means that we have a total of 2*4 or 8 combinations that yield the correct F & L terms.
We will see that some of these are ruled out! Remember that this polynomial does not have a gcf!
Factors of 15
Factors of 8
Factors of 6
Factors of 8
**Must both be
Negative!
Factors of 20
Factors of -9
**Must be
Opposites!
Factors of 12
Factors of 2
Factors of 8
Factors of 6
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