1. 14
Factoring and Applications

2. 14.1 Factors; The Greatest Common Factor
Objectives
1. Find the greatest common factor of a list of numbers. 2. Find the greatest common factor of a list of variable terms. 3. Factor out the greatest common factor. 4. Factor by grouping. 2

3. Find the Greatest Common Factor of a List of Numbers
The greatest common factor (GCF) of a list of integers is the largest common factor of those integers. This means 6 is the greatest common factor of 18 and 24, since it is the largest of their common factors.
Note
Factors of a number are also divisors of the number. The greatest common factor is the same as the greatest common divisor.

4. Find the Greatest Common Factor of a List of Numbers
Example 1
Find the greatest common factor for each list of numbers.
(a) 36, 60
First write each number in prime factored form.
36 = 2 · 2 · 3 · 3
60 = 2 · 2 · 3 · 5
Use each prime the least number of times it appears in all the factored forms. Here, the factored forms share two 2’s and one 3. Thus,
GCF = 2 · 2 · 3 = 12.

5. Find the Greatest Common Factor of a List of Numbers
Example 1 (continued)
Find the greatest common factor for each list of numbers.
(b) 18, 90, 126
Find the prime factored form of each number.
18 = 2 · 3 · 3
90 = 2 · 3 · 3 · 5
126 = 2 · 3 · 3 · 7
All factored forms share one 2 and two 3’s. Thus,
GCF = 2 · 3 · 3 = 18.

6. Find the Greatest Common Factor of a List of Numbers
Example 1 (concluded)
Find the greatest common factor for each list of numbers.
(c) 48, 61, 72
48 = 2 · 2 · 2 · 2 · 3
61 = 1 · 61
72 = 2 · 2 · 2 · 3 · 3
There are no primes common to all three numbers, so the GCF is 1.
GCF = 1

7. Find the Greatest Common Factor for Variable Terms
Note
The exponent on a variable in the GCF is the least exponent that appears on that variable in all the terms.
Example 2
Find the greatest common factor for each list of terms.
(a) 12x2, –30x5
12x2 = 2 · 2 · 3 · x2
–30x5 = –1 · 2 · 3 · 5 · x5
First, 6 is the GCF of 12 and –30. The least exponent on x is 2 (x5 = x2 · x3). Thus,
GCF = 6x2.

8. Find the Greatest Common Factor for Variable Terms
Example 2 (concluded)
Find the greatest common factor for each list of terms.
(b) –x5y2, –x4y3, –x8y6, –x7
–x5y2, –x4y3, –x8y6, –x7
There is no y in the last term. So, y will not appear in the GCF. There is an x in each term, and 4 is the least exponent on x. Thus,
GCF = x4.
Note In a list of negative terms, sometimes a negative common factor is preferable (even though it is not the greatest common factor). In (b) above, we might prefer –x4 as the common factor.

9. Find the Greatest Common Factor for Variable Terms
Finding the Greatest Common Factor (GCF)
Step 1 Factor. Write each number in prime factored form.
Step 2 List common factors. List each prime number or each variable that is a factor of every term in the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest common factor.)
Step 3 Choose least exponents. Use as exponents on the common prime factors the least exponents from the prime factored forms.
Step 4 Multiply. Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest common factor is 1.

10. Factor Out the Greatest Common Factor
CAUTION
The polynomial 3m + 12 is not in factored form when written as the sum
3 · m + 3 · Not in factored form
The terms are factored, but the polynomial is not. The factored form of 3m + 12 is the product
3(m + 4) In factored form

11. Factor Out the Greatest Common Factor
Example 3
Factor out the greatest common factor.
(a) 24x5 – 40x3
= 8x3(3x2) – 8x3(5)
GCF = 8x3
= 8x3(3x2 – 5)
Note If the terms inside the parentheses still have a common factor, then you did not factor out the greatest common factor in the previous step.

12. Factor Out the Greatest Common Factor
Example 3 (concluded)
Factor out the greatest common factor.
(b) 4x6y4– 20x4y3 + x2y2
= x2y2(4x4y2) – x2y2(20x2y) + x2y2(1)
= x2y2(4x4y2 – 20x2y +1)
CAUTION
Be sure to include the 1 in a problem like Example 3(b). Check that the factored form can be multiplied out to give the original polynomial. 12

13. Factor Out the Greatest Common Factor
Example 4
Factor – 3x5 – 15x3 + 6x2.
– 3x5 – 15x3 + 6x2
= – 3x2(x3 + 5x – 2)
GCF = – 3x2
Note
Whenever we factor a polynomial in which the coefficient of the first term is negative, we will factor out the negative common factor, even if it is just –1. 13

14. Factor Out the Greatest Common Factor
Example 5
Factor out the greatest common factor.
w2(z4– 3) + 5(z4 – 3)
Here, the binomial z4 – 3 is the GCF.
w2(z4– 3) + 5(z4 – 3)
= (z4– 3)(w2 + 5)

15. Factor By Grouping
Example 6
Factor by grouping.
6x + 4xy – 10y – 15
If we leave the terms grouped as they are, we could try factoring out the GCF from each pair of terms.
6x + 4xy – 10y – 15
= 2x(3 + 2y) – 5(2y + 3)
This works, showing a common binomial of 2y + 3 in each term.
6x + 4xy – 10y – 15
= 2x(2y + 3) – 5(2y + 3)
= (2y + 3)(2x – 5)

16. Factor By Grouping CAUTION
Be careful with signs when grouping in a problem like Example 6. It is wise to check the factoring in the second step, before continuing. 16

17. Factoring a Polynomial with Four Terms by Grouping
Factor By Grouping
Factoring a Polynomial with Four Terms by Grouping
Step 1 Group terms. Collect the terms into two groups so that each group has a common factor.
Step 2 Factor within groups. Factor out the greatest common factor from each group.
Step 3 Factor the entire polynomial. Factor a common binomial factor from the results of Step 2.
Step 4 If necessary, rearrange terms. If Step 2 does not result in a common binomial factor, try a different grouping. 17

18. Factor By Grouping
Example 7
Factor by grouping.
10a2 – 12b + 15a – 8ab
Working as before, we get
10a2 – 12b + 15a – 8ab
= 2(5a2 – 6b) + a(15 – 8b)
This does not work. These two factored terms have no binomial in common. So, we will group another way.

19. Factor By Grouping
Example 7 (concluded)
Factor by grouping.
10a2 – 12b + 15a – 8ab
= 10a2 – 8ab + 15a – 12b
= 2a(5a – 4b) + 3(5a – 4b)
This works, showing a common binomial of 5a – 4b in each term. Thus,
10a2 – 12b + 15a – 8ab
= (5a – 4b)(2a + 3)