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 :
x^{2} + xt + ax + at
Solution :
= x^{2} + xt + ax + at
Group the terms such that the terms in a gorup have a common divisor.
= (x^{2} + 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 :
2y^{3} + 4y^{2} + y + 2
Solution :
= 2y^{3} + 4y^{2} + y + 2
Group the terms such that the terms in a gorup have a common divisor.
= (2y^{3} + 4y^{2}) + (y + 2)
The first group has the common divisor y^{2} and the second group has the common divisor 1.
Factor out y^{2} from the girst group and 1 from the second group.
= 2y^{2}(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)(2y^{2} + 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 :
3x^{3} + 3x^{2} - 6x - 6
Solution :
= 3x^{3} + 3x^{2} - 6x - 6
In the polynomial above, all the terms have the common divisor 3.
Factor out 3 from all the terms.
= 3[x^{3} + x^{2} - 2x - 2]
Inside the bracket, group the terms such that the terms in a group have a common divisor.
= 3[(x^{3} + x^{2}) + (-2x - 2)]
Inside the bracket, the first group has the common divisor x^{2} and the second group has the common divisor -2.
Factor out x^{2} from the girst group and -2 from the second group.
= 3[x^{2}(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)(x^{2} - 2)]
= 3(x + 1)(x^{2} - 2)
Example 7 :
2ax^{2} - cx^{2} + 6a - 3c
Solution :
= 2ax^{2} - cx^{2} + 6a - 3c
Group the terms such that the terms in a gorup have a common divisor.
= (2ax^{2} - cx^{2}) + (6a - 3c)
The first group has the common divisor x^{2} and the second group has the common divisor 3.
Factor out x^{2} from the girst group and 3 from the second group.
= x^{2}(2a - c) + 3(2a - c)
Factor out (2a - c) from the two groups.
= (2a - c)(x^{2} + 3)
Example 8 :
4a^{2} + 5ab - 10b - 8a
Solution :
= 4a^{2} + 5ab - 10b - 8a
Group the terms such that the terms in a gorup have a common divisor.
= 4a^{2} + 5ab - 8a - 10b
= (4a^{2} + 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 :
a^{2}x + 3a^{2}y - 9x - 27y
Solution :
= a^{2}x + 3a^{2}y - 9x - 27y
Group the terms such that the terms in a gorup have a common divisor.
= (a^{2}x + 3a^{2}y) + (-9x - 27y)
The first group has the common divisor a^{2} and the second group has the common divisor -9.
Factor out a^{2} from the girst group and -9 from the second group.
= a^{2}(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)(a^{2} - 9)
= (x + 3y)(a^{2} - 3^{2})
= (x + 3y)(a + 3)(a - 3)
Example 10 :
3x^{3}y^{2} - 9x^{2}y^{3} + 9x^{2}y - 27xy^{2}
Solution :
3x^{3}y^{2} - 9x^{2}y^{3} + 9x^{2}y - 27xy^{2}
In the polynomial above, all the terms have the common divisor 3xy.
Factor out 3xy from all the terms.
= 3xy[x^{2}y - 3xy^{2} + 3x - 9y]
Inside the bracket, group the terms such that the terms in a group have a common divisor.
= 3xy[(x^{2}y - 3xy^{2}) + (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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