Study Guide: Factoring Using the Distributive Property

These web-based activities were developed, during summer sessions from 1997 through 1999, by students at Mainland High School, located in Daytona Beach, FL. As part of Mainland's CCTT grant, the students were given permission by the book's publisher to convert the ancillary materials of the County's adopted Algebra 1 Textbook into html format.

The distributive property has been used to multiply a polynomial by a monomial. It can also be used to express a polynomial in factored form. Compare the two columns below.

Directions: Complete the following.

20m

4xy²

A special case of using the distributive property is called factoring by grouping.

In this method (usually involving a polynomial with four terms), it is necessary to first group the terms and factor each group separately. The remaining polynomial factors of each group should be the same. This allows the distributive proprty to be applied a second time with a polynomial as the common factor.

Example 1: Factor the given polynomial using grouping.	12ac + 21ad + 8bc + 14bd
Notice that the first two terms each contain a factor of 3a, while the last two terms each contain a factor of 2b.	(12ac + 21ad) + (8bc + 14bd)
Notice that the expression (4c + 7d) has been revealed as a common factor.	3a(4c +7d) + 2b(4c + 7d)
The final factored form is	(4c + 7d)(3a + 2b)
Example 2: Factor the given polynomial using grouping.	15x − 3xy + 4y − 20
	(15x − 3xy) + (4y − 20)
	3x(5 − y) + 4(y − 5)
The grouping is close, but not quite identical. (y − 5) and (5 − y) are additive inverses. So we need to factor −1 from one of the expressions.	3x(5 − y) − 4(5 − y)
The final factored form is	(5 − y)(3x − 4)

8. a + 8a²b − ab

a(1 + 8ab − b)

14. 9x² − 3x

3x(3x − 1)

16. 45s³ − 15s²

15s²(3s − 1)

17. 18b²a − 4ba + 7ab²

ab(25b − 4)

19. −x⁵ − 4x⁴ + 23x³ − x

x(−x⁴ − 4x³ + 23x² − 1)

44. 3my − ab + am − 3by

((m − b)(3y + a)

47. 12mx − 8m + 6rx − 4r

2(2m + r)(3x − 2)