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Chapter 1
The Distributive Property
Tip
ctice
Pra
How can I make problems with several positive (+)
and negative (–) signs less complicated?
Since childhood, we have been taught that by
sharing we can break down barriers. In a case of
life imitating math, you can see the sharing
principle at work in algebra through something
called the distributive property.
Using the distributive property, you can eliminate
a set of grouping symbols, such as parentheses ( ),
by “sharing,” or multiplying, each number and
variable within the parentheses by a number or
variable outside of the parentheses. For example,
if you have 5(4 + a), you multiply the number and
variable within the parentheses by 5 to evenly
Positive Numbers
distribute the 5 to everything inside
the parentheses. In this example,
5(4 + a) = (5 x 4) + (5 x a) = 20 + 5a.
Distributing a positive number or variable
is straightforward, but you have to take
care when distributing negative numbers or
variables. When you distribute a negative
number or variable, all of the terms inside
the parentheses change signs from positive
to negative and/or from negative to positive.
For example, –1(2 + x) becomes –2 – x.
Negative Numbers
2(6 + 3) = (2
x
6) + (2
x
–3(2 + 4) = (–3
3)
= 12 + 6
6(4 + 5 + 9) = (6
x
4) + (6
If a problem contains positive or negative signs which
are side by side, you can simplify the problem a little
bit. When you find two positive signs or two negative
signs next to one another, you can replace the signs
with a single plus (+) sign.
2) + (–3
x
4)
5) + (6
x
9)
–2(4 – 5) = (–2
= 24 + 30 + 54
x
4) – (–2
1) 2(3 – 5)
2) 3(7 – 4)
3) 5(1 + 2)
4) –1(10 + 4)
If you find two opposite signs (+ or –) side by side,
you can replace the signs with a single minus (–) sign.
6) –5(–1 – 4)
5) –2(3 – 2)
7 + (–3) = 7 – 3
9 – (+2) = 9 – 2
Variables with Exponents
3(2 + x ) = (3
x
2) + (3
x
3( x 2 + y 2 ) = (3
x)
x
5)
–4( x – y ) = (–4
x
x ) – (–4
= (–8) – (–10)
= –4 x – (–4 y )
= –8 + 10
= –4 x + 4 y
x 2 ) + (3
x
x
x 2 (7 – x 2 ) = ( x 2
y)
x
x
7) – (4
x
5)
–(3 + 2) = (–1
that surround numbers you
need to add or subtract, you can
multiply the number outside the
parentheses by each number
inside the parentheses.
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3) + (–1
x
2)
a (a + b + c ) = (a
1 To remove the
parentheses ( ) that
surround numbers
you need to add or
subtract, you can
multiply the negative
number outside the
parentheses by each
number inside the
parentheses.
• If you see a negative sign
(–) by itself outside the
parentheses, assume the
number in front of the
parentheses is –1. For
example –(3 + 2) equals
–1(3 + 2).
Note: When you multiply two
numbers with different signs
(+ or –), the result will be a
negative number. For
example, –3 x 2 equals – 6.
x
a ) + (a
1 To remove the
parentheses ( ) that
surround numbers
and variables you
need to add or
subtract, you can
multiply the number
or variable outside
the parentheses by
each number and
variable inside the
parentheses.
7) – ( x 2
x
x 2)
= 7x 2 – x 4
x
b ) + (a
x
• A variable is a letter, such
as x or y, which represents
an unknown number.
Note: When you multiply a
variable by itself, you can use
exponents to simplify the
problem. For example, a x a
equals a 2 . For more information
on exponents, see page 54.
x
x 2 ) + (– x 3
x
y 4)
= (– x 3+2 ) + (– x 3 y 4 )
c)
= a 2 + ab + ac
= (–3) + (–2)
= 28 – 20
1 To remove the parentheses ( )
x
y 2)
= 7 x 2 – x 2+2
– x 3 ( x 2 + y 4 ) = (– x 3
4(7 – 5) = (4
x
= 3x 2 + 3y 2
= 6 + 3x
= (–6) + (–12)
x
Simplify the following expressions by
using the distributive property. You
can check your answers on page 250.
3 + (+5) = 3 + 5
6 – (–3) = 6 + 3
Variables
x
Algebra Basics
= –x 5 – x 3y 4
1 To remove the
parentheses ( ) that
surround numbers and
variables with exponents
you need to add or
subtract, you can multiply
the number or variable
outside the parentheses
by each number and
variable inside the
parentheses.
Note: When multiplying
numbers or variables
with exponents that have
the same base, you can
add the exponents. For
example, x 3 x x 2 equals
x 3+2 , which equals x 5 .
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