The Distributive Property
The Distributive Property:
Multiply by a Monomial
The product of a and (b+c) is given by:
a( b + c ) = ab + ac
Every term inside the parentheses is multiplied by a.
Example: Simplify 2x(x – 9)
Area Method:
x
-9
2
2x 2x -18x
“Arrow” Method:
2 x  x  9   2x 18x
2
2 x  18x
2
Do NOT forget to answer
the question.
The Generic Rectangle
Distribute:
+2
x
(x + 4)(x + 2)
2x 8
2
x 4x
x
+4
These
represent the
same area.
They must be
equal.
Area as a Product:
 x  4 x  2
Area as a Sum:
x  6x  8
2
Therefore:
2
x

4
x

2

x
 6x  8



The Distributive Property:
Multiply with the Area Model
2
3 terms times 2 terms
Distribute: ( x - x + 3 )( x + 5)
A 3x2 box.
The box is
generic so
don’t worry
about size.
x2
-x
+3
x
x3
-x2
+3x
+5
+5x2
-5x
+15
x3 – x2 + 3x + 5x2 – 5x + 15 = x3 + 4x2 – 2x + 15
Notice: Each of the three terms in the first set of parentheses is
multiplied by each in the second set of parentheses.
The Distributive Property:
Arrow Method
2
Distribute: ( x - x + 3 )( x + 5)
Instead of the making a box, you can multiply each of the three
terms in the first set of parentheses by each in the second set of
parentheses.
x3 + 5x2 – x2 – 5x + 3x + 15 = x3 + 4x2 – 2x + 15
The Distributive Property:
FOIL
Write the following as a sum:
Multiply the…
•
•
•
•
•
( 3x – 2 )( 2x + 7)
Firsts
Outers 6x2 + 21x + -4x + -14
Inners
Lasts
= 6x2 + 17x – 14
Simplify
Mr. Wells considers FOIL to be an F-word. It can only be used in specific
instances. It only works for a binomial multiplied by a binomial. It is not
worth memorizing.
The Distributive Property and Solving
Equations
Solve:
-x
5  x  x  3    x  5 x  1
x
2
-x
+3
-3x
+1
x
5
x
x2
5x
x
+5
5  x  3x    x  6 x  5 
2
2
5  x  3x   x  6 x  5
5  3x  6 x  5
5  3x  5
3x  10
x  10 3
2
2