Use the distributive Property to Expand Algebraic Expressions

a, b and c being real numbers, The distributive property property is given by:

To understand the distributive property, let us evaluate the expression 2(3 + 4) using two ways.

1) Use of the order of operations:

2(3 + 4)

= 2(7) group what is inside the parentheses

= 14 multiply to simplify

2) Use of the distributive property

2(3 + 4)

= (2)(3) + (2)(4) distribute the 2 over the two terms inside the parentheses.

= 6 + 8 simplify

= 14 simplify

Both methods give the same answer. But if you have to simplify or expand an expression with a variable such as

2( x + 6)

x + 6 cannot be simplified because x is a variable. Hence the use of the distributive property.

2( x + 6)

= (2)(x) + (2)(6)

= 2x + 12

Only the distributive property can be used to expand 2(x + 6) because of the variable x.

Use the distributive Property to Factor Alebraic Expressions

The distributive propery can in general be written as

a( b + c) = a b + a c

The same property can be used from right to left to factor (write as a product) expressions such as a b + a c

Example: Factor (write as a product) the expression 3 x + 6

3 x + 6 Given

= 3 x + 3 (2) write 6 as 3 × 2 = 3(2)

= 3(x + 2) factor out the common factor 3

Answer the Following Questions

Solutions to the Above Questions

1. Solution
   Use the distributive property to expand the expressions.
   
   1. 2(x + 2) given
      = (2)(x) + (2)(2) use distributive property
      = 2 x + 4 simplify

      
      

   2. 3(a + 4) given
      = (3)(a) + (3)(4) use distributive property
      = 3 a + 12 simplify

      
      

   3. 4(3 + b) given
      = (4)(3) +(4)(b) use distributive property
      = 12 + 4 b simplify

      
      

   4. 5(3 + n) given
      = (5)(3) + (5)(n) use distributive property
      = 15 + 5 n simplify

      
      

   5. 2(a + b) given
      = (2)(a) + (2)(b) use distributive property
      = 2 a + 2 b simplify

      
      

   6. 2(x + y + 4) given
      = (2)(x) + (2)(y) +(2)(4) use distributive property
      = 2 x + 2 y + 8 simplify

      
      

   7. (7 + b) 4 given
      = (7)(4) + (b)(4) use distributive property
      = 28 + 4 b simplify
2. Solution
   Expand and simplify.
   
   
   1. 3(x + 1) + 3 given
      = (3)(x) +(3)(1) + 3 use distributive property to expand
      = 3 x + 3 + 3 simplify
      = 3 x + 6 simplify

      
      

   2. 5(1 + n) + 6 given
      = (5)(1) + (5)(n) + 6 expand
      = 5 + 5 n + 6 simplify
      = 5 n + 11 simplify

      
      

   3. 5(a + 2) + 2(a + 3) given
      = (5)(a) +(5)(2) + (2)(a) + (2)(3) use distributive property to expand
      = 5 a + 10 + 2 a + 6 simplify
      = (5 a + 2 a) + (10 + 6) group like terms
      = 7 a + 16 simplify

      
      

   4. 2(1 + b) + 6(b + 2) + 4 given
      = (2)(1) + (2)(b) + (6)(b) + (6)(2) + 4 use distributive property to expand
      = 2 + 2 b + 6 b + 12 + 4 simplify
      = (2 b + 6 b) + (2 + 12 + 4) group like terms
      = 8 b + 18 simplify

      
      

   5. 2(a + b) + 3 (a + b) given
      = (2)(a) + (2)(b) + (3)(a) + (3)(b) use distributive property to expand
      = 2 a + 2 b + 3 a + 3 b simplify
      = (2 a + 3 a) + (2 b + 3 b) group like terms
      = 5 a + 5 b simplify
   
   
3. Solution
   We first need to find a common factor and then factor using distributive property from right to left as follows
   a b + a c = a (b + c )
   1. 2 x + 4 given
      = 2 (x) + 2(2) Find a common factor to the terms in the given expression; in this case it is 2
      = 2(x + 2) factor 2 out

      
      

   2. 3 x + 3 given
      = 3(x) + 3(1) Find a common factor to the terms in the given expression; it is 3
      = 3 (x + 1) factor 3 out

      
      

   3. 4 a + 12 given
      = 4(a) + 4(3) Find a common factor to the terms in the expression; it is 4
      = 4(a + 3) factor 4 out

      
      

   4. 21 + 7 b given
      = 7(3) + 7(b) Find a common factor to the terms in the expression; it is 7
      = 7(3 + b) factor 7 out

      
      

   5. 15 + 5 x given
      = 5(3) + 5(x) Find a common factor ; it is 5
      = 5(3 + x) factor 5 out

      
      

   6. x / 2 + 1 / 2 given
      = (1/2) (x) + (1/2)(1) Find a common factor ; it is 1/2
      = (1/2)(x + 1) factor 1/2 out