FACTORING POLYNOMIALS BY GROUPING

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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