With these
three points in
mind,
the
question
is:
How can
we use real
numbers in
practical
calculations?
What rules
apply?
Distributive
Property . .
.
The
Distributive
Property
combines
the
following
two
operations
in
mathematics:
multiplication
and
addition
. It is
formally
defined as
"the
distribution
of
multiplication
over
addition".
There
is
only
one
Distributive
Property
which
combines
(both)
addition
and
multiplication.
There is
not a
separate
Distributive
Property
for
addition,
and
another
Distributive
Property
for
multiplication.
The
Distributive
Property
allows us
to
simplify
an
expression
containing
parentheses
to an
expression
without
parentheses.
As an
example
, the
Distributive
Property
can be
used to
multiply
the
following
sum
of
two
numbers
by
3
:
5 + 6
This problem
would
ordinarily be
written as
expression
which contains
parentheses:
3
*
(
5 + 6
)
To evaluate the
expression, you
could add the 5
and the 6
together first,
and then
multiply by
3
5 + 6 =
11
3
*
11 =
33
However, you
can also
multiply each
of the two
numbers by 3
separately, and
then add the
results
together.
3
*
5 =
15
3
*
6 =
18
15
+
18 =
33
The
results are the
same
either way.
Simply put, the
Distributive
Property
states
that it makes
no difference
whether
you:

multiply 3
times each
number
individually,
then add the
result
or
 add
the two numbers
together, then
multiply the
result by 3
The
Distributive
Law states that
the
following
expressions
are
equivalent
to one
another:
3
*
(
5 + 6
)
= 3
*
5 +
3
*
6 =
33
Mental
Math
 the
Distributive
Law simplifies
calculations
Knowing
that the two
expressions
shown above are
equivalent is
very useful
when
multiplying
numbers using
mental
math.
For
example,
multiply the
following two
numbers using
mental
math:
5
*
723
Using
the
Distributive
Law, this
number 723 can
be arbitrarily
spit into
parts, and
rewritten as
shown below.
This is called
Decomposing
the
Multiplier
.
5
*
723
= 5
* (
700 + 20
+ 3
)
The
multiplication
process can be
completed as
follows:
5
*
( 700 + 20
+ 3 ) =
5 *
700
+ 5
*
20
+ 5
*
3
= 3500 +
100 + 15
=
3615
Algebra
 the
Distributive
Law helps
simplify
algebraic
expressions
The
Distributive
Law is even
more important
in Algebra. The
values of the
variables used
in Algebra are
unknown. The
Distributive
Law allows you
to simplify an
algebraic
expression
without knowing
the actual
values
represented by
the
variables.
The
Distributive
Law can be
stated using
algebraic
notation as
follows:
a
*
(
b + c
)
= a
*
b +
a *
c
If more
than three
variables are
used, the
Distributive
Law can be
stated:
a
*
(
b + c + d + e +
... + z
)
= a
*
b +
a *
c +
a *
d +
a *
e + ... +
a
*
z
Example
Problems in
Algebra using
the
Distributive
Law
:
Example
Problems Using
the
Horizontal
Format
:
Example
1:
3
(
x + 9
)
= 3
*
(
x + 9
)
=
3
*
x +
3 *
9
=
3x +
27
Example
2:
r
(
x + 9
)
= r
*
(
x + 9
)
=
r
*
x +
r *
9
=
rx +
9r
Example
Problems Using
a
Multiplication
Table
:
Example
1:
3
(
x + 9
)
(+3)
(
+ x +
9
)


+x 

+9 
+3 

(+3)
*
(+x) 

(+3)
*
(+9) 
Multiplication
Table
(+3)
(
+ x +
9
)
= [
(+3)
*
(+x)
] + [
(+3)
*
(+9)
]
=
3x +
27
Example
2:
(
x + 2
)
(
x  5
)
(
+ x +
2
)
(
+ x 
5
)


+x 

5 

+x 

(+x)
*
(+x) 

(+x)
*
(5) 
+2 

(+2)
*
(+x) 

(+2)
*
(5) 
Multiplication
Table
= [
x
*
x
]  [
x
*
5
] + [
2
*
x
]  [
2
*
5
]
=
x^{2}
 5x + 2x 
10
=
x^{2}
+ [ 5x + 2x] 
10
=
x^{2}
 3x 
10
Example
Problems Using
the
Vertical
Format
:
Example
1:
(
x + 2
)
(
x  5
)
(
+ x +
2
)
(
+ x 
5
)
+
x −
5
x
+
x +
2
2 x
−
10
(+2)
*
(
+ x 
5
)
x^{2}
−
5
x
(+x)
*
(
+ x 
5
)
_____________
x^{2}
−
3 x
− 10
Example
2:
(
x + 2
)
(
x^{2} +
x  5
)
(
+ x +
2
)
(
+
x^{2}
+
x −
5
)
+
x^{2}
+
x
−
5
x
+
x
+
2
+ 2
x^{2}
+
2 x
−
1 0
(+2)
*
(
+
x^{2}
+
x −
5
)
+
x^{3}
+
x^{2} −
5 x
(+x)
*
(
+
x^{2}
+
x −
5
)
_______________________
x^{3} +
3
x^{2}
−
3 x
−
1
0