Section 6.3 Factoring Trinomials
Factoring by ‘ac-method’ is used for trinomials and some binomials
 Factoring ac-method will be discussed with examples
Examples for positive ac
1. Factor the given polynomial
7
10
2. Factor the given polynomial
5
4
Step 1: Factor out GCF. GCF = 1, so DONE!
Step 1: Factor out GCF. GCF = 1, so DONE!
Step 2: Find a, b, c
Step 2: Find a, b, c
1,
7,
10
1,
5,
4
Step 3: Find ac
Step 3: Find ac
1 ∙ 10 10
1∙4 4
Step 4: Find positive factors of ac whose products
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
is ac and whose sum is ‘b’
Positive factors of 10: 1 10 , 2 5
Positive factors of 4: 1 4 , 2 2
Since ac is positive, there are two case;
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 1: Both factors are positive
Case 2: Both factors are negative
Case 2: Both factors are negative
Since ‘b’ is positive, both factors are positive
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
Thus, possible factors are
1 10 ,
, 2
2
Step 5: Rewrite the middle term, bx, using the two
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
numbers found in step 4 as coefficients. Then
factoring by grouping.
factoring by grouping.
7
10
10
5
4
4
2
5
10
4
4
5
4
5
4
3. Factor the given polynomial 4
22
10
4. Factor the given polynomial 8
36
28
Step 1: Factor out GCF. GCF = 2
Step 1: Factor out GCF. GCF = 4
4
22
10 2 2
11
5
8
36
28 4 2
9
7
Step 2: Find a, b, c
Step 2: Find a, b, c
2,
11,
5
2,
9,
7
Step 3: Find ac
Step 3: Find ac
2 ∙ 5 10
2 ∙ 7 14
Step 4: Find positive factors of ac whose products
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
is ac and whose sum is ‘b’
Positive factors of 10: 1 10 , 2 5
Positive factors of 14: 1 14 , 2 7
Since ac is positive, there are two case;
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 1: Both factors are positive
Case 2: Both factors are negative
Case 2: Both factors are negative
Since ‘b’ is positive, both factors are positive
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
Thus, possible factors are
, 2 5
1
14 ,
Step 5: Rewrite the middle term, bx, using the two
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
numbers found in step 4 as coefficients.
Then, factoring by grouping.
Then, factoring by grouping.
2 2
11
5
2 2
5
4 2
9
7
4 2
7
2 2
10
5
4 2
2
7
7
2
5
42
7
2
5
4
2
7
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 1
Section 6.3 Factoring Trinomials
Examples for negative ac
1. Factor the given polynomial
3
10
2. Factor the given polynomial
3
10
Step 1: Factor out GCF. GCF = 1, so DONE!
Step 1: Factor out GCF. GCF = 1, so DONE!
Step 2: Find a, b, c
Step 2: Find a, b, c
1,
3,
10
1,
3,
10
Step 3: Find ac
Step 3: Find ac
1 ∙ 10
10
1 ∙ 10
10
Step 4: Find positive factors of ac whose products
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
is ac and whose sum is ‘b’
Positive factors of 10: 1 10 , 2 5
Positive factors of 10: 1 10 , 2 5
Since ac is negative, one factor is positive
Since ac is negative, one factor is positive
and the other is negative
and the other is negative
Since ‘b’ is positive, bigger factor is positive
Since ‘b’ is negative, bigger factors is negative
Thus, possible factors are
Thus, possible factors are
1
10,
1
10 ,
Step 5: Rewrite the middle term, bx, using the two
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
numbers found in step 4 as coefficients. Then
factoring by grouping.
factoring by grouping.
3
10
10
3
10
10
2
5
10
2
5
10
5
5
5
5
3. Factor the given polynomial 4
22
10
4. Factor the given polynomial 6
15
21
Step 1: Factor out GCF. GCF = 2
Step 1: Factor out GCF. GCF = 3
4
22
10 2 2
9
5
6
27
21 3 2
5
7
Step 2: Find a, b, c
Step 2: Find a, b, c
2,
11,
5
2,
5,
7
Step 3: Find ac
Step 3: Find ac
2∙ 5
10
2∙ 7
14
Step 4: Find positive factors of ac whose products
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
is ac and whose sum is ‘b’
Positive factors of 10: 1 10 , 2 5
Positive factors of 14: 1 14 , 2 7
Since ac is negative, one factor is positive
Since ac is negative, one factor is positive
and the other is negative
and the other is negative
Since ‘b’ is positive, bigger factor is positive
Since ‘b’ is negative, bigger factors is negative
Thus, possible factors are
Thus, possible factors are
, 2
5
1
14 ,
Step 5: Rewrite the middle term, bx, using the two
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
numbers found in step 4 as coefficients.
Then, factoring by grouping.
Then, factoring by grouping.
2 2
11
5
2 2
5
3 2
5
7
3 2
7
2 2
10
5
3 2
2
7
7
2
5
32
7
2
5
3
2
7
Note: The polynomials are NOT FACTORABLE if no combination of factors in the step 4 is
found to have whose sum is the middle term ‘b’.
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 2
Section 6.3 Factoring Trinomials
Exercises
(Solution 1)
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
1,
10,
21
Step 3: Find ac
1 ∙ 21 21
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 21: 1 21 , 3 7
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is positive, both factors are positive
Thus, possible factors are 1 21 ,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
factoring by grouping.
10
21
21
3
7
21
7
7
(Solution 2)
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
1,
13,
40
Step 3: Find ac
1 ∙ 40 40
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 40:
1 40 , 2 20 , 4 10 , 5 8
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
1
40 , 2
20 ,
4
10 ,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
factoring by grouping.
13
40
5
40
5
8
40
8
8
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 3
Section 6.3 Factoring Trinomials
(Solution 3)
Step 1: Factor out GCF. GCF = 1, so DONE!
Step 2: Find a, b, c
1,
11,
26
Step 3: Find ac
1 ∙ 26
26
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 26: 1 26 , 2 13
Since ac is negative, one factor is positive
and the other is negative
Since ‘b’ is positive, bigger factors is positive
Thus, possible factors are 1
26,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
Then, factoring by grouping.
11
26
26
2
13
26
13
13
(Solution 4)
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
2,
11,
5
Step 3: Find ac
2 ∙ 5 10
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 10: 1 10 , 2 5
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is positive, both factors are positive
Thus, possible factors are
, 2 5
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
Then, factoring by grouping.
2
11
5 2
5
2
10
5
5
5
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 4
Section 6.3 Factoring Trinomials
(Solution 5) Ignore FOIL multiplication
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
7,
50,
7
Step 3: Find ac
7 ∙ 7 49
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 49: 1 49 , 7 7
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
, 7
7 ,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
factoring by grouping.
50
7 7
7
7
7
49
7
7
7
Ignore FOIL multiplication
(Solution 6) Ignore FOIL multiplication
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
6,
5,
3
Step 3: Find ac
6 ∙ 3 18
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 18: 1 18 , 2 9 , 3 6
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
1
18 , 2
9 , 3
6
1
18
19
2
9
11
3
6
9
No combination gives the middle term – 5.
Thus, the polynomial is prime.
Ignore FOIL multiplication
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 5
Section 6.3 Factoring Trinomials
(Solution 7) Ignore FOIL multiplication
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
4,
12,
9
Step 3: Find ac
4 ∙ 9 36
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 36:
1 36 , 2 18 , 3 12 , 4 9 , 6 6
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
1
36 , 2
18 , 3
12
4
9 ,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
factoring by grouping.
4
12
9 4
9
4
6
6
9
2
3
2
3
Ignore FOIL multiplication
(Solution 8)
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
3,
10,
32
Step 3: Find ac
3 ∙ 32
96
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 96:
1 96 , 2 48 , 3 32 ,
4 24 , 6 16 , 8 12
Since ac is negative, one factor is positive and
the other is negative
Since ‘b’ is negative, the bigger is negative
Thus, possible factors are
1
96 , 2
48 , 3
32 ,
4
24 ,
,8
12
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
Then, factoring by grouping.
3
10
32 3
32
3
6
16
32
3
16
3
16
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 6
Section 6.3 Factoring Trinomials
10
(Solution 9)
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
10,
31,
14
Step 3: Find ac
10 ∙ 14
140
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 140:
1 140 , 2 70 , 4 35 ,
5 28 , 7 20 , 10 14
Since ac is negative, one factor is positive and
the other is negative
Since ‘b’ is positive, the bigger is positive
Thus, possible factors are
1
140, 2
70,
,
5
28, 7
20, 10
14
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
Then, factoring by grouping.
31
14
10
14
10
4
35
14
2
7
2
7
(Solution 10)
Step 1: Factor out GCF. GCF is 2
12
52
10
6
31
5
Step 2: Find a, b, c
6,
31,
5
Step 3: Find ac
6 ∙ 5 30
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 30:
1 30 , 2 15 , 3 10 , 5 6
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is positive, both factors are positive
Thus, possible factors are
, 2 15 , 3 10 , 5 6
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
Then, factoring by grouping.
2 6
31
5
2 6
5
2 6
30
5
2
5
2
5
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 7
Section 6.3 Factoring Trinomials
(Solution 11)
Step 1: Factor out GCF. GCF = 4y
28
256
36
7
64
9
Step 2: Find a, b, c
7,
64,
9
Step 3: Find ac
7∙9
63
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 63:
1
63 , 3
21 , 7
9
Since ac is positive, there are two case;
Case 1: Both factors are positive
Case 2: Both factors are negative
Since ‘b’ is negative, both factors are negative
Thus, possible factors are
,
3
21 ,
7
9
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
factoring by grouping.
4 7
64
9
4 7
4 7
4
4
9
63
9
9
9
(Solution 12)
Step 1: Factor out GCF. GCF =
32
44
8
11
10
Step 2: Find a, b, c
8,
11,
10
Step 3: Find ac
8 ∙ 10
80
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 80:
1
80 , 2
40 , 4
20 , 5
16 , 8
10
Since ac is negative, one factor is positive and
the other is negative
Since ‘b’ is positive, the bigger is positive
Thus, possible factors are
1
80,
,
2
8
40,
10
4
20,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients.
Then, factoring by grouping.
4
Cheon-Sig Lee
8
11
www.coastalbend.edu/lee
10
4
4
4
4
8
8
or 4
8
5
5
16
10
10
2
2
2
Page 8