No. Odd numbers of negative integers will have a negative product. Even numbers of negative integers will have a positive product.
Examples
-1 x -3 x -4 = -12
-3 x -4 = 12
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Yes. Only by multiplying a positive by a negative, will the product be negative.
It may be either. If any of the integers is zero, the product will be zero. Else, if one or three of the integers is negative, the product will be negative. Otherwise, it will be positive.
Oh, dude, let me break it down for you. If you multiply three integers, the product will be positive if all three integers are either positive or negative. If you have two negative integers and one positive integer, the product will be negative. So, like, it depends on the mix of positives and negatives, you know?
The product of three positive numbers is positive. The product of three negative numbers is negative.
The first three positive integers, 1, 2, and 3, satisfy this condition.
The product of two integers cannot be "positive and negative".