Factoring Using the Distributive Property
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. |
|
a
20m
1
4xy2
|
Notice that the missing term is the GCF between the two monomials.
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) |
Directions: Factor each selected polynomial.
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) |