Question

Is the product of an even number of factors always a positive number?

Short answer

Yes, the product of an even number of factors is always a positive number, assuming that no factor equals zero.

Step-by-step solution

Analyze the properties of multiplication

Before beginning, it is critical to recall the primary properties of multiplication concerning positive and negative values. The product of two positive numbers or two negative numbers is always a positive number. On the other hand, the product of one positive and one negative number is a negative number.

Apply multiplication properties to an even number of factors

Let's consider an even number of factors. This implies having two or more factors for multiplication. If all factors are positive, the product will certainly be positive since the multiplication of any number of positive factors will always yield a positive product, irrespective of the total count of the factors.

Consider the case of negative factors

Now, let's consider the scenario of negative factors. If there is an even number of negative factors, the product will also be positive. This is because a negative factor multiplied by a negative becomes positive, and the sequence continues. Thus, even if all factors are negative, as long as their count is even, the product is positive.

Consider mixed positive and negative factors

In a situation where there are both positive and negative factors, their count does not affect the sign of the product, so long as the total number of negative factors is even. As a negative factor multiplied by a negative results in a positive, the overall product remains positive if there is an even number of negative factors. In case of an odd number of negative factors, the product becomes negative.