Download Factoring Trinomials The Expansion Method

Factoring Trinomials
The Expansion Method
The polynomials that we work with most often are trinomials. Recall
that trinomials are three-term polynomials. They are in the form of:
ax2 + bx + c, where a and b are the numbers in front of x2 and x,
respectively, and c is the constant at the end. We will look at three types
of trinomials. This section will focus on the trinomials where a ≠ 1.
There are two main ways to factor trinomials where a ≠ 1. One is
commonly referred to as the trial and error method. This method seems
to be rather difficult for many students and less reliable. Thus, we will
focus on the second way which is called the expansion method
(sometimes called the ac method). While the expansion method can be a
little time consuming, it is very accurate. With the expansion method,
we multiply the first and last numbers (a and c), and then select the two
factors when added equal the middle number (b). Those two factors will
be used to rewrite and expand the middle term.
Example 1: Factor: 3x2 + 11x + 6
Multiply first and last numbers: 3· 6 = 18
Factors of 18: 1,18; 2,9; 3,6
That add to get 11: 1 + 18 = 19 No
2 + 9 = 11 Yes, use these two numbers
3x2 + 2x + 9x + 6
Rewrite and expand 11x
2
3x + 2x + 9x + 6
Split the polynomial in half
x(3x + 2) + 3(3x + 2) Factor the GCF from each side
(3x + 2)(x + 3)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
We could have also expanded the 11x as 9x + 2x
3x2 + 9x + 2x + 6
Split the polynomial in half
3x(x + 3) + 2(x + 3) Factor the GCF from each side
(x + 3)(3x + 2)
Same two binomial factors
To check the answer, multiply: (x + 3)(3x + 2)
x(3x + 2) + 3(3x + 2)
3x2 + 2x + 9x + 6
3x2 + 11x + 6 Correct
Example 2: Factor: 8y2 – 2y – 15
Multiply first and last numbers: 8 · (-15) = -120
Factors of 120: 1,120; 2,60; 3,40; 4, 30; 5,24; 6,20; 8,15; 10,12
Add to get -2. Since the product is -120, one number must be negative:
-1+120 = 119 or -120+1 = -119 No
-2+60 = 58 or -60+2 = -58
No
-3+40 = 37 or -40+3= -37
No
-4+30 = 26 or -30+4 = -26
No
-5+24 = 19 or -24+5 = -19
No
-6+20 = 14 or -20+6 = -14
No
-8+15 = 7 or -15+8 = -7
No
-10+12 = 2 or -12+10 = -2
Yes
8y2 – 12y + 10y – 15
8y2 – 12y + 10y – 15
4y(2y – 3) + 5(2y – 3)
(2y – 3)(4y + 5)
Rewrite and expand -2y
Split the polynomial in half
Factor the GCF from each side
To check the answer, multiply: (2y – 3)(4y + 5)
2y(4y + 5) – 3(4y + 5)
8y2 + 10y – 12y – 15
8y2 – 2y – 15
Correct
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 3: Factor: 4x2 – xy – 5y2
Multiply the first and last numbers: 4 · (-5) = -20
Factors of 20: 1,20; 2,10; 4,5
Add to get -1. Since the product is -20, one number must be negative:
-1+20 = 19 or -20+1 = -19 No
-2+10 = 8 or -10+2 = -8 No
-4+5 = 1 or -5+4 = -1
Yes
4x2 – 5xy + 4xy – 5y2 Rewrite and expand –xy
4x2 – 5xy + 4xy – 5y2 Split the polynomial in half
x(4x – 5y) + y(4x – 5y) Factor the GCF from each side
(4x – 5y)(x + y)
To check the answer, multiply: (4x – 5y)(x + y)
4x(x + y) – 5y(x + y)
4x2 + 4xy – 5xy – 5y2
4x2 – xy – 5y2 Correct
When factoring, it is important not to forget about the GCF. If all terms
have a common factor, first factor out the GCF before factoring further.
Example 4: Factor: 18x3 + 33x2 – 30x
3x(6x2 + 11x – 10)
Factor out the GCF, 3x, first
Multiply the first and last numbers: 6 · (-10) = -60
Factors of 60: 1,60; 2,30; 3,20; 4,15; 5,12; 6,10
Add to get 11. Since the product is -60, one number must be negative:
-1+60 = 59 or -60+1 = -59 No
-2+30 = 28 or -30+2 = -28 No
-3+20 = 17 or -20+3 = -17 No
-4+15 = 11 or -15+4 = -11 Yes
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
3x(6x2 – 4x + 15x – 10) Rewrite and expand 11x
3x (6x2 – 4x + 15x – 10) Split the polynomial in half
3x[2x(3x – 2) + 5(3x – 2)] Factor the GCF from each side
3x(3x – 2)(2x + 5)
To check the answer, multiply: 3x(3x – 2)(2x + 5)
3x[3x(2x + 5) – 2(2x + 5)]
3x[6x2 + 15x – 4x – 10]
3x[6x2 + 11x – 10]
18x3 + 33x2 – 30x Correct
Example 5: Factor: 3z2 + 2z – 7
Multiply the first and last numbers: 3 · (-7) = -21
Factors of 21: 1,21; 3,7
Add to get 2. Since the product is -21, one number must be negative:
-1+21 = 20 or -21+1 = -20 No
-3+7 = 4 or -7+3 = -4
No
Since no set of factors add to 2, the trinomial cannot be factored.
Prime
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)