Even or Odd Numbers of Negative Factors Handout
PA_M2_S2_T3
Many times I multiply more than two integers and I might encounter numerous factors
that are negative. So I want to talk about having two or more negative factors and
performing multiplication with those. Here is an example where I have four factors.
I have -3 times 2 times -4 times -2.
Step 1: I'm going to do the
first two as a product, -3
times positive 2 is minus 6.
That's the first multiplication
problem I come to in my
expression.
Step 2: Then I'm going to
multiply that times -4, and
Step 3: finally I'll multiply
that by -2.
-6 times -4, well I have two negatives,
the same sign here, which means I'm going
to get a positive. 6 times 4 is 24 and
because I have two negatives, it becomes
positive 24. And then I multiply that,
finally, times my last negative factor,
-2. The product of 24 and -2 is -48.
Let's look at these two values, too:
This is an exponential and
remember that that's just
repeated multiplication, so that
in the first case where I have -2
raised to the third power, that's
the same thing as saying -2 times
-2 times -2.
Let's see what that gives us. The
first two, as I'd take my
multiplication in order from left
to right, is -2 times -2. That
product is positive 4. Finally, I
do this last multiplication,
positive 4 times -2 gives me 8.Because I have different signs
here my product of these two
integers will be -8.
Let's look at what happens with -3 when I raise it to the fourth power.
Remember that this is the same thing
is -3 times -3 times -3 times -3, and
again will take my multiplication in
order from left to right.
-3 times -3 gives me positive 9.
Now I multiply 9 times my -3 right
here, that's a negative and a
positive, it's going to have a
negative sign and it's going to be 27 times -3.
I finally complete my multiplication
-27 times -3 is positive 81. If I
have two integers that are the same
sign the product will be positive,
and 27 times 3 is 81.
Notice that in the first case (
), I had an odd number of negative
factors, and in the second case (
) I had an even number of negative
factors. So we can make a very interesting observation based on that. If I
have an odd number of negative factors the product will be negative. An
even number of negative factors, the product will be positive. This is a
very important observation and it can save you a lot of time if you pay
attention to the number of negative factors you have in any product,
because you'll know based on whether it's even or odd whether are your
result will be positive or negative.