Distributing multiplication over addition (and subtraction) and factoring (the opposite of distributing) are extremely important in algebra. The distributive law of multiplication over addition, a(b + c) = (ab + ac), says that you can first take the sum (b + c) then the product (a times the sum of b and c) or the individual products (ab and ac) then the sum (the sum of ab and ac). For instance, 12(6 + 4) could be computed as 12(6 + 4) = 12(6) + 12(4) = 72 + 48 = 120 or as 12(6 + 4) = 12(10) = 120. The distributive law of multiplication over subtraction, a(b – c) = ab – ac, says the same about a product and difference.
Practice
Solutions
Sometimes you will need to
‘‘distribute’’ a minus sign or negative sign: –(a + b) = –a – b and –(a – b) =
–a + b. You can
use the distributive properties
and think of –(a + b) as (–1)(a + b)
and –(a –
b) as (–1)(a
– b) :
–(a + b) = (–1)(a + b)
= (–1)a + (–1)b = –a + –b = –a – b
and
–(a – b) = (–1)(a – b)
= (–1)a – (–1)b = –a – (–1)b
= –a – (– b) = = –a + b
A common mistake is to write –(a +
b) = –a + b and
–(a – b) = –a – b. The minus sign and negative sign in front of the parentheses
changes the signs of
every term (a quantity separated by a plus or minus sign) inside the parentheses.
Practice
1.
– (4 + x)
=
2.
– (–x –
y) =
3.
– (x2
– 5x – 6) =
Solutions
1.
– (4 + x)
= –4 – x
2.
– (–x –
y) = x + y
3.
– (x2
– 5x – 6) = x2 + 5x + 6
Distributing negative quantities has
the same effect on signs as distributing a minus sign: every sign in the parentheses changes.
Practice
1.
– 2 (16
+ y) =
2.
– 50 (3 –
x) =
3.
– 7x2
(– x – 4y) =
Solutions
1.
– 2 (16
+ y) = –32 – 2y
2.
– 50 (3 –
x) = –150 + 50x
3.
– 7x2
(– x – 4y) = 7x3 + 28x2y
Sumber
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