Download F2 - Sum of Cubes

2
Factoring by Grouping
0.1
Warm-up, “X” Game
Begin this lesson by playing the game, but this time with a product that has
even more factors and sums.
Warm Up. Begin by creating a table of Factors and Sums for 30 and −30.
Below, the tables are started for you; however, you should try to complete the
tables yourself. Completed tables are provided at the end of the lesson.
Table of
Product
30
30
..
.
Factors and Sums
Factors Sum
=
1 · 30
31
= −1 · −30 −31
..
..
=
.
.
Table of
Product
−30
−30
..
.
Factors and Sums
Factors Sum
= −1 · 30
29
= 1 · −30
−29
..
..
.
.
=
Use the tables you created to complete the following games. Try the games
before checking the solutions at the end of the lesson.
30
30
30
−30
31
11
−13
−29
−30
−30
30
30
7
−1
13
−17
−30
−30
−30
30
29
−13
1
17
1
0.2
30
30
−30
−30
−31
−11
13
−7
Factoring by Grouping
Consider ax + ay + bx + by. No GCF is shared by all four terms, i.e., the GCF
is 1. Look at the “groups” of two terms that do share a GCF. The first two
terms share a GCF of a, and the second group of two terms share a GCF of b.
1. Consider the polynomial as two groups of two.
2. Factor the common monomial for each group.
3. The new GCF will be a binomial.
a is the gcf
z }| {
ax + ay + bx + by = ax + ay + bx + by
| {z }
b is the gcf
= a(x + y) + b(x + y), (x + y) is the gcf.
= (x + y)(a + b)
Notice the use of the under brace to denote the groups, not parentheses. Avoid
using parentheses to show the groups, for in some cases this changes the expression.
Is the following expression true or false? Justify your answer using the properties
of operations.
(x + y)(a + b) = (a + b)(x + y)
The above is true because of the commutative property of multiplication1 , i.e.,
the order of the factors does not affect the multiplication. An example of the
commutative property is that 3 · 4 = 4 · 3.
Example. Factor. 3xy − 4y − 4 + 3x.
1 The commutative property of multiplication is often shortened to the commutative property. When this is done, the reader can determine by the context of the step whether the
author is referring to the commutative property of multiplication of the commutative property
of addition.
2
Solution. The groups of two do not have an obvious GCF; however,
the commutative property allows for terms to be rearranged.
3xy − 4y − 4 + 3x = 3xy + 3x − 4y − 4 Commutative property
= 3x(y + 1) − 4(y + 1) The GCF of each group of two.
= (y + 1)(3x − 4), y + 1 is the GCF
OR,
3xy − 4y − 4 + 3x = y(3x − 4) + 3x − 4, commutative property
= y(3x − 4) + 1(3x − 4), the second group has a GCF=1
= (3x − 4)(y + 1)
Solutions to games and Table of Factors and Sums for 30 and −30.
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
30
1
30
30
31
5
Sums
Sum
29
−29
13
−13
7
−7
1
−1
30
6
−3
11
−10
−13
3
−30
1
−30
−29
−30
10
−3
5
30
3
−6
10
−2
−15
7
−1
13
−17
−30
−30
−30
30
30
−1
2
6
−5
−15
2
15
29
−13
1
17
30
30
−30
−30
−1
−30
−5
−31
0.3
30
−30
−6
15
−2
13
−11
3
−10
−7
Practice Problems
Directions. Factor the expressions by grouping.
1.
xy + 5x + 2y + 10
2.
xy − 4x + 3y − 12
3.
6xy − 4x + 9y − 6
4.
15xy − 10x + 6y − 4
5.
3xy − 12x + 2y − 8
6.
ab − 7a + 4b − 28
3
3
2
7.
x + 5x + 3x + 15
8.
x − 6x2 + 2x − 12
9.
xy + 4x − 3y − 12
10.
xy + 5y − 2x − 10
11.
xy − 3x − 5y + 15
12.
2xy − 8x − 3y + 12
13
xy + 3y + x + 3
14.
xy − 5y + x − 5
15.
3xy − 6x + y − 2
16.
2xy − 3y − 2x + 3
17.
xy + 10 + 5x + 2y
18.
xy − 12 − 4x + 3y
19.
6xy − 6 − 4x + 9y
20.
3xy − 4 + 3x − 4y
21.
xy + 2x + 14 + 7y
22.
xy + 3x + 15 + 5y
23.
2x + xy + 4y + 8
24.
3x + xy + 7y + 21
25.
2ab − 5a + 6b − 15
26.
6xy − 4x − 9y + 6
4
27.
3ab + 15 + 9a + 5b
28.
29.
8ab − 4a − 6b + 3
30. 2xy + 14x − 21 − 3y
31.
6ab − 3b + 2a − 1
32.
33. 14b + 21 + 3a + 2ab
2xy + 4x + y + 2
8xy + 2x + 12y + 3
34. 12xy − 3 + 18x − 2y
5
Solutions
1.
xy + 5x + 2y + 10 = (y + 5)(x + 2)
2.
xy − 4x + 3y − 12 = (y − 4)(x + 3)
3.
5.
6xy − 4x + 9y − 6 = (3y − 2)(2x + 3)
4.
15xy − 10x + 6y − 4 = (3y − 2)(5x + 2)
3xy − 12x + 2y − 8 = (y − 4)(3x + 2)
6.
3
2
2
7.
x + 5x + 3x + 15 = (x + 5)(x + 3)
9.
xy + 4x − 3y − 12 = (y + 4)(x − 3)
8.
ab − 7a + 4b − 28 = (b − 7)(a + 4)
3
x − 6x2 + 2x − 12 = (x − 6)(x2 + 2)
10.
xy + 5y − 2x − 10 = (x + 5)(y − 2)
11.
xy − 3x − 5y + 15 = (y − 3)(x − 5)
12.
2xy − 8x − 3y + 12 = (y − 4)(2x − 3)
13
xy + 3y + x + 3 = (x + 3)(y + 1)
14.
xy − 5y + x − 5 = (x − 5)(y + 1)
15.
3xy − 6x + y − 2 = (y − 2)(3x + 1)
16.
2xy − 3y − 2x + 3 = (2x − 3)(y − 1)
17.
xy + 10 + 5x + 2y = (y + 5)(x + 2)
18.
xy − 12 − 4x + 3y = (y − 4)(x + 3)
19.
6xy − 6 − 4x + 9y = (3y − 2)(2x + 3)
20.
3xy − 4 + 3x − 4y = (y + 1)(3x − 4)
21.
xy + 2x + 14 + 7y = (y + 2)(x + 7)
22.
xy + 3x + 15 + 5y = (y + 3)(x + 5)
23.
2x + xy + 4y + 8 = (y + 2)(x + 4)
24.
3x + xy + 7y + 21 = (y + 3)(x + 7)
25.
2ab − 5a + 6b − 15 = (2b − 5)(a + 3)
26.
6xy − 4x − 9y + 6 = (3y − 2)(2x − 3)
27.
3ab + 15 + 9a + 5b = (b + 3)(3a + 5)
28.
2xy + 4x + y + 2 = (y + 2)(2x + 1)
29.
8ab − 4a − 6b + 3 = (2b − 1)(4a − 3)
30.
2xy + 14x − 21 − 3y = (y + 7)(2x − 3)
31.
6ab − 3b + 2a − 1 = (2a − 1)(3b + 1)
32.
8xy + 2x + 12y + 3 = (4y + 1)(2x + 3)
33.
14b + 21 + 3a + 2ab = (2b + 3)(a + 7)
34.
12xy − 3 + 18x − 2y = (2y + 3)(6x − 1)
6