Title: Algebra 1
Author: Mandy McDaniel
Last saved by: Glencoe Mathematics

Lesson 8-2

Example 1 Use the Distributive Property
Use the Distributive Property to factor each polynomial.
a.	21xy – 18x
First, find the GCF of 21xy and 18x^2.
21xy = 3 ⋅ 7 ⋅ x ⋅ y	Factor each monomial.
18x= 2 ⋅ 3 ⋅ 3 ⋅ x ⋅ x	Circle the common prime factors. 
GCF: 3 ⋅ x or 3x
Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF.
21xy – 18x 	= 3x (7 ⋅ y) – 3x(2 ⋅ 3 ⋅ x)	Rewrite each term using the GCF.
	= 3x (7y) – 3x (6x)	Simplify remaining factors. 
	= 3x (7y – 6x)	Distributive Property
Thus, the completely factored form of 21xy – 18x is 3x (7y – 6x).

b.	64rs – 32rs + 8rs
64rs = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ r ⋅ r ⋅ r ⋅ s	Factor each monomial.
 32rs= 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 	 r ⋅ r ⋅ s ⋅ s ⋅ s	Circle the common prime factors.
  8rs= 2 ⋅ 2 ⋅ 2 ⋅ 	 r ⋅ r ⋅ s ⋅ s	
GCF: 2 ⋅ 2 ⋅ 2 ⋅ r ⋅ r ⋅ s or 8rs. 
64rs – 32rs + 8rs
	= 8rs (8r) – 8rs (4s) + 8rs(s)	Rewrite each term using the GCF.
	= 8rs(8r – 4s + s)	Distributive Property

Example 2 Use Grouping
Factor 2x^2+ x – 8x – 4.
	2x+ x – 8x – 4
		= (2x+ x) + (– 8x – 4)	Group terms with common factors.
		= x (2x + 1) + -4(2x + 1)	Factor the GCF from each grouping.
		= (2x + 1)(x + -4)	Distributive Property
		= (2x + 1)(x – 4)	Simplify.

Example 3 Use the Additive Inverse Property
Factor 3a^2 – 3a + 4 – 4a.
3a– 3a + 4 – 4a
	= (3a – 3a) + (4 – 4a)	Group terms with common factors.
	= 3a(a – 1) + 4(1 – a)	Factor the GCF from each grouping.
	= 3a(a – 1) +4(-1)(a – 1)	1– a = -1(a – 1)
	= 3a(a – 1) +-4(a – 1)	4(-1) = -4
	= (a – 1)(3a + -4)	Distributive Property
	= (a – 1)(3a – 4)	Simplify.

Example 4 Solve an Equation 
a.   Solve (q + 2)(2q – 1) = 0. Check the solutions.
If (q + 2)(2q – 1) = 0, then according to the Zero Product Property either q + 2 = 0 or 2q – 1 = 0.

  (q + 2)(2q – 1)	= 0	Original equation
	q + 2	= 0 	or 	  2q – 1 	= 0	Set each factor equal to zero.
		q = -2			 2q	= 1	Solve each equation.
				   q 	= 
The roots are –2 and .
Check: Substitute –2 and  for q in the original equation.
		 (q + 2)(2q – 1) 	= 0		 (q + 2)(2q – 1) 	= 0
		  (-2 + 2)[2(-2) – 1] 	 0		( + 2)[2() – 1]  0
		 (0)(-5) 	 0	()(0) 	 0
		   0 	= 0  	0 	= 0  

b.   Solve 3x+ 9x = 0. Check the solutions.
Write the equation so that it is of the form ab = 0.
		3x+ 9x	= 0	Original equation.
		 3x(x + 3)	= 0	Factor the GCF of 3x^2 and 9x, which is 3x.
		 3x	= 0	or 	x + 3 = 0	Zero Product Property
		   x	= 0		x = -3	Solve each equation.
The roots are –3 and 0.