# FACTORING POLYNOMIALS BY GROUPING

The following steps would be useful to factor polynomials by grouping.

Step 1 :

Arrange the terms such that two or more twrems have a common divisor.

Step 2 :

For each pair, factor out the greatest common divisor.

Step 3 :

Find the greatest common divisor of the groups and factor it out.

Factor each of the following polynomials :

Example 1 :

ax + bx + ay + by

Solution :

= ax + bx + ay + by

Group the terms such that the terms in a gorup have a common divisor.

= (ax + bx) + (ay + by)

The first group has the common divisor x and the second group has the common divisor y.

Factor out x from the girst group and y from the second group.

= x(a + b) + y(a + b)

The two groups above have the common divisor (a + b).

Factor out (a + b) from the two groups.

= (a + b)(x + y)

Example 2 :

x2 + xt + ax + at

Solution :

= x2 + xt + ax + at

Group the terms such that the terms in a gorup have a common divisor.

= (x2 + xt) + (ax + at)

The first group has the common divisor x and the second group has the common divisor a.

Factor out x from the girst group and a from the second group.

= x(x + t) + a(x + t)

The two groups above have the common divisor (x + t).

Factor out (x + t) from the two groups.

= (x + t)(x + a)

Example 3 :

2y3 + 4y2 + y + 2

Solution :

= 2y3 + 4y2 + y + 2

Group the terms such that the terms in a gorup have a common divisor.

= (2y3 + 4y2) + (y + 2)

The first group has the common divisor y2 and the second group has the common divisor 1.

Factor out y2 from the girst group and 1 from the second group.

= 2y2(y + 2) + 1(y + 2)

The two groups above have the common divisor (y + 2).

Factor out (y + 2) from the two groups.

= (y + 2)(2y2 + 1)

Example 4 :

xy - 5y - 2x + 10

Solution :

= xy - 5y - 2x + 10

Group the terms such that the terms in a gorup have a common divisor.

= (xy - 5y) + (-2x + 10)

The first group has the common divisor y and the second group has the common divisor -2.

Factor out y from the girst group and -2 from the second group.

= y(x - 5) - 2(x - 5)

The two groups above have the common divisor (x - 5).

Factor out (x - 5) from the two groups.

= (x - 5)(y - 2)

Example 5 :

6ax + 3bd - 2ad - 9bx

Solution :

= 6ax + 3bd - 2ad - 9bx

Group the terms such that the terms in a gorup have a common divisor.

6ax - 9bx - 2ad + 3bd

= (6ax - 9bx) + (-2ad + 3bd)

The first group has the common divisor 3x and the second group has the common divisor -d.

Factor out 3x from the girst group and -d from the second group.

= 3x(2a - 3b) - d(2a - 3b)

The two groups above have the common divisor (2a - 3b).

Factor out (2a - 3b) from the two groups.

= (2a - 3b)(3x - d)

Example 6 :

3x3 + 3x2 - 6x - 6

Solution :

= 3x3 + 3x2 - 6x - 6

In the polynomial above, all the terms have the common divisor 3.

Factor out 3 from all the terms.

= 3[x3 + x2 - 2x - 2]

Inside the bracket, group the terms such that the terms in a group have a common divisor.

= 3[(x3 + x2) + (-2x - 2)]

Inside the bracket, the first group has the common divisor x2 and the second group has the common divisor -2.

Factor out x2 from the girst group and -2 from the second group.

= 3[x2(x + 1) - 2(x + 1)]

Inside the bracket, the two groups have the common divisor (x + 1).

Factor out (x + 1) from the two groups.

= 3[(x + 1)(x2 - 2)]

= 3(x + 1)(x2 - 2)

Example 7 :

2ax2 - cx2 + 6a - 3c

Solution :

= 2ax2 - cx2 + 6a - 3c

Group the terms such that the terms in a gorup have a common divisor.

= (2ax2 - cx2) + (6a - 3c)

The first group has the common divisor x2 and the second group has the common divisor 3.

Factor out x2 from the girst group and 3 from the second group.

= x2(2a - c) + 3(2a - c)

Factor out (2a - c) from the two groups.

= (2a - c)(x2 + 3)

Example 8 :

4a2 + 5ab - 10b - 8a

Solution :

= 4a2 + 5ab - 10b - 8a

Group the terms such that the terms in a gorup have a common divisor.

4a2 + 5ab - 8a - 10b

= (4a2 + 5ab) + (-8a - 10b)

The first group has the common divisor a and the second group has the common divisor -2.

Factor out a from the girst group and -2 from the second group.

= a(4a + 5b) - 2(4a + 5b)

The two groups above have the common divisor (4a + 5b).

Factor out (4a + 5b) from the two groups.

= (4a + 5b)(a - 2)

Example 9 :

a2x + 3a2y - 9x - 27y

Solution :

= a2x + 3a2y - 9x - 27y

Group the terms such that the terms in a gorup have a common divisor.

= (a2x + 3a2y) + (-9x - 27y)

The first group has the common divisor a2 and the second group has the common divisor -9.

Factor out a2 from the girst group and -9 from the second group.

= a2(x + 3y) - 9(x + 3y)

The two groups above have the common divisor (x - 5).

Factor out (x - 5) from the two groups.

= (x + 3y)(a2 - 9)

= (x + 3y)(a2 - 32)

= (x + 3y)(a + 3)(a - 3)

Example 10 :

3x3y2 - 9x2y3 + 9x2y - 27xy2

Solution :

3x3y2 - 9x2y3 + 9x2y - 27xy2

In the polynomial above, all the terms have the common divisor 3xy.

Factor out 3xy from all the terms.

= 3xy[x2y - 3xy2 + 3x - 9y]

Inside the bracket, group the terms such that the terms in a group have a common divisor.

= 3xy[(x2y - 3xy2) + (3x - 9y)]

Inside the bracket, the first group has the common divisor xy and the second group has the common divisor 3.

Factor out xy from the first group and 3 from the second group.

= 3xy[xy(x - 3y) + 3(x - 3y)]

Inside the bracket, the two groups have the common divisor (x - 3y).

Factor (x - 3y) from the two groups.

= 3xy[(x - 3y)(xy + 3)]

3xy(x - 3y)(xy + 3)

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