Recall the rules of multiplication of real numbers and think of possible combinations you can make with different values.
False Counterexample: (-2)(2)(2)
Practice makes perfect
We need to tell if the statement shown below is true. Otherwise, we are asked for a counterexample.
If the product of three numbers is negative,
then all the numbers are negative.
When multiplying two real numbers with the same sign, we obtain a positive result. Conversely, if factors have different signs, we obtain a negative number as the result.
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Same Sign & Different Signs
(+)(+)=(+) & (+)(-)=(-)
(-)(-)=(+) & (-)(+)=(-)
To get a negative result having 3 factors, we need the product of two of them to have a different sign than the remaining factor. This happens if all of them are negative, just as the statement says.
(-2)(-2)(-2) = 4(-2)= -8
However, this is not the only way. Let's first try multiplying a negative and a positive number, for example (-2)(2) = -4. This gives a negative number as the result. Then, to obtain a negative one more time, we can multiply it by a positive number. Putting it all together, we have another 3 factors that would make it work.
(-2) (2 )( 2) = (-4)(2) = - 8
As we can see, multiplying a negative number by two other positive numbers will give a negative result as well. Thus, the statement is false.
