Question

6 how many integers must be negative for the product of three integers to be negative? explain. find two different groups of three integers whose product is −12. show your work.

Explanation:

Step1: Determine negative integer count

The product of integers has a sign based on the number of negative factors. The rule is: the product of an odd number of negative integers is negative, and the product of an even number (including zero) of negative integers is positive. For three integers, an odd number of negative integers means 1 or 3 negative integers (since 1 and 3 are the odd numbers between 1 and 3).

Step2: Find first group of three integers

We need three integers whose product is -12. Let's start with -1, 3, and 4. Multiply them: $(-1) \times 3 \times 4 = -12$.

Step3: Find second group of three integers

Let's take -2, 2, and 3. Multiply them: $(-2) \times 2 \times 3 = -12$.

Answer:

To have the product of three integers negative, 1 or 3 integers must be negative (because the product of an odd number of negative numbers is negative). Two groups of three integers with product -12 are: \((-1, 3, 4)\) (since \((-1)\times3\times4 = -12\)) and \((-2, 2, 3)\) (since \((-2)\times2\times3 = -12\)).