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PRACTICE WITH PRODUCTS OF SIGNED VARIABLES
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The concepts for this exercise are summarized below.
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PRODUCTS INVOLVING SIGNED VARIABLES
The product (-a)b
can be written in a variety of ways:
(-a)b
⏞
the opposite of a, times b
=
a(-b)
⏞
a, times the opposite of b
=
-ab
⏞
the opposite of ab
In all these cases,
the same three numbers are being multiplied ( -1 ,
a , and b ),
so the result is the same.
USE OF PARENTHESES AND CENTERED DOTS
People have different preferences as to how much or how little they use parentheses,
and how much or how little they use the centered dot to denote multiplication.
There are places where parentheses and centered dots are needed,
but in other situations they're optional.
You might see any of these:
ab = (a)b = a(b) = (a)(b) =
a · b = a · (b) = (a) · b = (a) · (b)
JUXTAPOSITION OF VARIABLES DENOTES MULTIPLICATION
When two variables are juxtaposed (i.e., sitting next to each other),
as in the expression ab ,
the operation between them is multiplication.
Thus, ab is the simplest (and preferred) way to denote a
multiplied by b .
WITH SIGNED VARIABLES, PARENTHESES ARE USUALLY REQUIRED
When a signed variable is involved, parentheses are usually required.
To multiply a by -b (in that order)
we can't just juxtapose them and write a-b ,
because this would look like subtraction.
Thus, we must write a(-b) .
EVEN AND ODD NUMBERS OF FACTORS OF NEGATIVE ONE
When you see opposites in multiplication problems,
just treat the minus sign as a factor of -1 , and recall these rules:
Any even number of factors of -1 is positive:
for example, (-1)(-1)(-1)(-1) = 1 .
Any odd number of factors of -1 is negative:
for example, (-1)(-1)(-1) = -1 .
Conventionally, the minus sign (if there is one) is pulled to the front of the expression,
as illustrated in these examples:
(-a)(-b) = ab (two factors of -1 )
-(-a)(b) = ab (two factors of -1 )
-a(-b) = ab (two factors of -1 )
-(-a)(-b) = -ab (three factors of -1 )
(-a)(-b)(-c)(d) = -abcd (three factors of -1 )
EXAMPLES:
Is -a(-b) a name for ab ? Answer: YES
Is -b(-1)a a name for -ab ? Answer: NO