Download Section 6.5 A General Factoring Strategy

Section 6.5 A General Factoring Strategy
Difference of Two Squares:

 OTE: Sum of Two Squares,
, is not factorable
Sum and Differences of Two Cubes:


Perfect Square Trinomials:

2
2

Strategy for Factoring Polynomials
 Step 1: Factor out GCF, if possible
 Step 2: Determine the number of terms in the polynomial
 Step 3: Determine a factoring technique as follows:
If the given polynomial is a Binomial, factoring by one of the following
1.
2.
3.
If the given polynomial is a Trinomial, factoring by ac-method
If the given polynomial has 4 or more terms, factoring by Grouping
Exercises
(Solution 1)
Step 1: Factor out GCF. GCF = 2x
2
8
2
4
Since
4 is the combination of two linear,
the polynomial is factored completely.
Factoring is done!
The answer is 2
4 or 2
4
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 1
Section 6.5 A General Factoring Strategy
(Solution 2)
Step 1: Factor out GCF. GCF = 1
Step 2: Determine the number of terms: Two terms
Step 3: Determine a factoring technique
Since two terms, power 2, and difference,
for factoring
we are using
Factoring by
Step 1: Rewrite in terms of perfect squares
4
25 2
5
2
5
Step 2: Find a and b, then factor out with
⟹
2 ,
5
Two linear combination, so factoring is done
(Solution 3)
Step 1: Factor out GCF. GCF = 1
Step 2: Determine the number of terms: Two terms
Step 3: Determine a factoring technique
Since two terms, power 3, and difference,
we are using
for factoring
Factoring by
Step 1: Rewrite in terms of perfect squares
216
1 6
1
6
1
Step 2: Find a and b, then factor out with
6 ,
1
1 ⟹
6
1 36
6
1
(Solution 4)
2
2
is four terms, so By Grouping
Factoring by Grouping
Step 1: Group first two terms and last two terms
2
2
Step 2: Factor out GCF from each group
GCF for the first group is 2
GCF for the second group is x
2
2
2
2
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 2
Section 6.5 A General Factoring Strategy
(Solution 5) Ignore FOIL multiplication
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
8,
22,
15
Step 3: Find ac
8 ∙ 15 120
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 40:
1 120 , 2 60 , 3 40 , 4 30
5 24 , 6 20 , 8 15 , 10 12
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
120 , 2
60 , 3
40
4
30 , 5
24 , 6
20
8
15 ,
Step 5: Rewrite the middle term, bx, using the two
numbers found in step 4 as coefficients. Then
factoring by grouping.
8
22
15 8
15
8
15
2
3
2
3
Ignore FOIL multiplication
(Solution 6)
Step 1: Factor out GCF. GCF = 3x
3
12
4
4
2 ⟹ Sum of Two Squares
The sum of two squares is not factorable
So, factoring is done!
The answer is 3
4
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 3
Section 6.5 A General Factoring Strategy
(Solution 7)
Step 1: Factor out GCF. GCF = 5x
10
5
2
1
Since 5 2
1 is the combination of two linear,
the polynomial is factored completely.
Factoring is done!
The answer is 5 2
1
(Solution 8)
Step 1: Factor out GCF. GCF is 1, so DONE!
Step 2: Find a, b, c
1,
5,
25
Step 3: Find ac
1 ∙ 25 25
Step 4: Find positive factors of ac whose products
is ac and whose sum is ‘b’
Positive factors of 25: 1 25 , 5 5
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
25 , 5
5
1
25
26
5
5
10
No combination gives the middle term – 5.
Thus, the polynomial is prime.
(Solution 9)
Step 1: Factor out GCF. GCF = 3x
3
27
9
Step 2: Determine the number of terms:
9 has two terms
Step 3: Determine a factoring technique
Since two terms, power 2, and difference,
we are using
for factoring
Factoring by
Step 1: Rewrite in terms of perfect squares
3
9
3
Step 2: Find a and b, then factor out with
⟹
,
3
3
3
Three linear combination, so factoring is done
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 4
Section 6.5 A General Factoring Strategy
(Solution 10)
Step 1: Factor out GCF. GCF = 1
Step 2: Determine the number of terms: Two terms
Step 3: Determine a factoring technique
Since two terms, power 2, and sum, it is prime
36
6 ⟹
Remember!
Sum of Two Squares is not factorable
Cheon-Sig Lee
www.coastalbend.edu/lee
Page 5