Download 1 The Greatest Common Factor

1
The Greatest Common Factor
0.1
The ”X” Game
When two numbers are multiplied together, the result is called the product
and the two numbers are called the factors. For example, in 6 = 2 · 3, the 6
is the product, the 2 and the 3 are the factors. A simple game is based on
finding factors of numbers that sum, or add, to a specific number. The game
is displayed as an ”X” with the product in the top position and the sum in the
bottom position. The answers can be any integers and are written on the sides
of the ”X”. Play the following game.
6
5
This game is played by finding two numbers that multiply to 6 and sum to 5.
The answers are 2 and 3 because 2·3 = 6 and 2+3 = 5. The game’s presentation
is then written
6
2
3
5
The game can quickly become more interesting when negative products are used.
Try the game
−6
−5
After some pondering, you will find the answers are
1
−6
1
−6
−5
The game serves as a way to record or direct one to find two numbers with both
a certain sum and product. This will become the key to factoring second degree
polynomials. Initially this process may seem a bit confusing. Try writing all
the possible factors of 6 and the associated sums. Then try the same with −6.
You should do this on your own first, then compare your table with the tables
below.
Table of
Product
6
6
6
6
Factors and Sums
Factors Sum
=
1·6
7
= −1 · −6
−7
=
2·3
5
= −2 · −3
−5
Table of
Product
−6
−6
−6
−6
Factors and Sums
Factors Sum
= −1 · 6
5
= 1 · −6
−5
= −2 · 3
1
= 2 · −3
−1
Use the table to complete the following games. Try the games before checking
the solutions.
6
6
−6
7
−5
1
6
−6
−6
−7
5
−1
Note that 6 = 2 · 3 = 3 · 2 by the commutative property. Since these are the
same regardless of order, only record one or the other as possible factors of 6.
2
Practice. Construct a Table of Factors and Sums for 30 and −30. Please
try this yourself by modeling the above examples before checking the solutions.
Be systematic in your process of finding all possible factors. It is the act of
personal construction and investment of time that results in learning.
Solutions to games and Table of Factors and Sums for 30 and −30.
6
1
6
6
−2
−6
−3
3
−2
7
−5
1
6
−6
−6
−1
−6
−7
6
−1
5
2
−3
−1
Table of
Product
30
30
30
30
30
30
30
30
Factors and Sums
Factors Sum
=
1 · 30
31
=
2 · 15
17
=
3 · 10
13
=
5·6
11
= −1 · −30 −31
= −2 · −15 −17
= −3 · −10 −13
= −5 · −6 −11
Table of
Product
−30
−30
−30
−30
−30
−30
−30
−30
Factors and
Factors
= −1 · 30
= 1 · −30
= −2 · 15
= 2 · −15
= −3 · 10
= 3 · −10
= −5 · 6
= 5 · −6
3
Sums
Sum
29
−29
13
−13
7
−7
1
−1
0.2
The Greatest Common Factor
The distributive property explains how a sum can be written as a product and
vicea versa. The distributive property is
a(x + y) = ax + ay
The left side of the identity is a product, and the right side of the identity is a
sum. When the distributive property is used to write a product as a sum, this
is called distributing or multiplying. When the distributive property is used to
write a sum as a product, this is referred to as factoring. The a is called the
greatest common factor and is often abbreviated as GCF. The a represents the
greatest factor common to each term. The above example of the distributive
property is the simplest form. The concept extends to three, four or even many
terms inside the parentheses.
a(x1 + x2 + x3 + . . . + xn ) = ax1 + ax2 + ax3 + . . . + axn , ∀n ∈ N
a(x + y)
−→
←−
=
ax + ay
Distributing
Factoring
When two or more numbers are multiplied, the result is a product. In order to
factor a polynomial, begin by finding the greatest factor that is common to all
terms, greatest common factor (GCF).
Example. Factor the sum 12x3 y + 9x2 y 2 . This means write the sum as a
product. The key is to think in terms of multiplication. What is the largest
factor common to all terms?
12x3 y + 9x2 y 2 = 3x2 y · 4x + 3x2 y · 3y The GCF is 3x2 y.
= 3x2 y(4x + 3y)
The GCF is placed outside the parentheses.
Example. Factor the sum 3xy−x. This means write the sum as a product. The
key is to think in terms of multiplication. What is the largest factor common
to all terms?
3xy − x = x(3y − 1) The GCF is x.
The GCF is placed outside the parentheses.
4
What is most important about this example is that the entire second term was
the GCF. When the GCF is factored out, notice that a −1 still remains. A
common mistake is to drop the entire term, but in factoring, each term should
always have a place holder. When the GCF is factored from a binomial, a
binomial should still remain in the parentheses. If a polynomial has three terms,
after the GCF is factored, the polynomial in the parentheses should still have
three terms, and so forth.
Example. Factor the sum 4x2 + 12x + 20. This means write the sum as a
product. The key is to think in terms of multiplication. What is the largest
factor common to all terms?
4x2 + 12x + 20 = 4 · x2 + 4 · 3x + 4 · 5 The GCF is 4.
= 4(x2 + 3x + 5)
The GCF is placed outside the parentheses.
Example. Factor the sum 5a + xa.
In this binomial the GCF is a. To factor this the GCF is placed outside the
parentheses. The answer is
5a + xa = a(5 + x).
Example. Factor the sum 5(a) + x(a).
This problem intentionally looks similar to the above example. In this binomial
the GCF is (a). To factor this the GCF is placed outside the parentheses. The
answer is
5a + xa = (a)(5 + x).
Example. Factor the sum 5(a + h) + x(a + h). This problem intentionally looks
similar to the above example above, but it is different. In the earlier version the
GCF was simply (a). In this example, the GCF is the entire binomial contained
in the parentheses, (a + h).
Thus,
5(a + h) + x(a + h) = (a + h)(5 + x).
0.3
Practice Problems
Find the greatest common factor (GCF).
5
1.
3.
5.
7.
9.
11.
3x + 6
12x + 8
2x2 + 7x
9x2 − 6x
8x3 y 2 + 12x2 y
27x3 y − 36x2
2.
4.
6.
8.
10.
12.
6x − 4
15x − 6
5y 2 − 2y
15y 2 + 10y
6x4 y 3 − 9x2 y
45x3 y 2 + 60x3 yz
Practice Problems. Factor out the the greatest common factor (GCF).
13.
15.
17.
19.
21.
23.
25.
27.
29.
31.
33.
35.
37.
5x + 10
8x + 4
12y 2 − 18y + 24
10a2 − 5a + 15
2x2 + 3x
15x2 − 10x
6x2 + 2x
2x2 y + 10xy − 6y
12x2 y − 6xy + 3y
10x4 y 3 + 5x3 y 2 − 15x3
9x2 y 2 + 18x2 y − 27xy 2
24a2 b2 + 16a2 b − 8ab
20x2 y − 15xy 2 + 5x
14.
16.
18.
20.
22.
24.
26.
28.
30.
32.
34.
36.
38.
6x − 9
2x2 + 8x − 6
14a2 + 7a − 21
15x − 5
2y 2 − 5y
12x2 + 8x
6x2 − 3x
15ab2 − 10ab + 20a
10a2 b2 − 4a2 b + 2ab
12x3 y 2 − 24x2 y 2 − 36y 2
42x3 y 2 + 7x2 y 2 − 14xy 2
30a4 b3 − 45a3 b2 + 15a2 b2
6x2 y 2 + 18x2 y − 6y
Practice Problems Factor out the binomial factor.
39.
41.
43.
45.
47.
a(x + 1) + 3(x + 1)
x(a − 5) − 4(a − 5)
x(x − 2) + 3(x − 2)
x(y + 1) + (y + 1)
2x(y + 3) − (y + 3)
40.
42.
44.
46.
48.
6
b(x + 4) − 2(x + 4)
x(b + 3) + y(b + 3)
x(x + 7) − 2(x + 7)
x(a + 2) − (a + 2)
5a(x − 2) + (x − 2)
Solutions to Practice Problems.
Find the greatest common factor (GCF).
1.
3.
5.
7.
9.
11.
3x + 6 the GCF is 3
12x + 8 the GCF is 4
2x2 + 7x the GCF is x
9x2 − 6x the GCF is 3x
8x3 y 2 + 12x2 y the GCF is 4x2 y
27x3 y − 36x2 the GCF is 9x2
2.
4.
6.
8.
10.
12.
6x − 4 the GCF is 2
15x − 6 the GCF is 3
5y 2 − 2y the GCF is y
15y 2 + 10y the GCF is 5y
6x4 y 3 − 9x2 y the GCF is 3x2 y
45x3 y 2 + 60x3 yz the GCF is 15x3 y
Practice Problems. Factor out the the greatest common factor (GCF).
13.
15.
17.
19.
21.
23.
25.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
5x + 10 = 5(x + 2)
8x + 4 = 4(2x + 1)
12y 2 − 18y + 24 = 6(2y 2 − 3y + 4)
10a2 − 5a + 15 = 5(2a2 − a + 3)
2x2 + 3x = x(2x + 3)
15x2 − 10x = 5x(3x − 2)
6x2 + 2x = 2x(3x + 1)
2x2 y + 10xy − 6y = 2y(x2 + 5x − 3)
15ab2 − 10ab + 20a = 5a(3b2 − 2b + 4)
12x2 y − 6xy + 3y = 3y(4x2 − 2x + 1)
10a2 b2 − 4a2 b + 2ab = 2ab(5ab − 2a + 1)
10x4 y 3 + 5x3 y 2 − 15x3 = 5x3 (2xy 3 + y 2 − 3)
12x3 y 2 − 24x2 y 2 − 36y 2 = 12y 2 (x3 − 2x2 − 3)
9x2 y 2 + 18x2 y − 27xy 2 = 9xy(xy + 2x − 3y)
42x3 y 2 + 7x2 y 2 − 14xy 2 = 7xy 2 (6x2 + x − 2)
24a2 b2 + 16a2 b − 8ab = 8ab(3ab + 2a − 1)
30a4 b3 − 45a3 b2 + 15a2 b2 = 15a2 b2 (2a2 b − 3a + 1)
20x2 y − 15xy 2 + 5x = 5x(4xy − 3y 2 + 1)
6x2 y 2 + 18x2 y − 6y = 6y(x2 y + 3x2 − 1)
14.
16.
18.
20.
22.
24.
26.
6x − 9 = 3(2x − 3)
2x2 + 8x − 6 = 2(x2 + 4x − 3)
14a2 + 7a − 21 = 7(2a2 + a − 3)
15x − 5 = 5(3x − 1)
2y 2 − 5y = y(2y − 5)
12x2 + 8x = 4x(3x + 2)
6x2 − 3x = 3x(2x − 1)
Practice Problems Factor out the binomial factor.
39.
41.
43.
45.
47.
a(x + 1) + 3(x + 1) = (x + 1)(a + 3)
x(a − 5) − 4(a − 5) = (a − 5)(x − 4)
x(x − 2) + 3(x − 2) = (x − 2)(x + 3)
x(y + 1) + (y + 1) = (y + 1)(x + 1)
2x(y + 3) − (y + 3) = (y + 3)(2x − 1)
7
40.
42.
44.
46.
48.
b(x + 4) − 2(x + 4) = (x + 4)(b − 2)
x(b + 3) + y(b + 3) = (b + 3)(x + y)
x(x + 7) − 2(x + 7) = (x + 7)(x − 2)
x(a + 2) − (a + 2) = (a + 2)(x − 1)
5a(x − 2) + (x − 2) = (x − 2)(5a + 1)