Final answer:
The product of a positive number and a negative number is always negative, regardless of their magnitudes.we can confidently say that whenever you multiply a positive number by a negative number, the product will be negative.
Step-by-step explanation:
The conjecture that describes the product of a positive number and a negative number is, it is always negative. This is because when two numbers with opposite signs are multiplied together, the product will have a negative sign. For example, multiplying a positive number by a negative number, like (-3) x 2, results in a negative product, which is -6. Similarly, 4 x (-4) gives us -16.
These rules apply regardless of the magnitude of the numbers involved, meaning the size of the numbers does not affect the sign of the product.
In mathematics, a conjecture is an educated guess that is based on some observations. Let's make a conjecture about the product of a positive number and a negative number by examining the rules of signs for multiplication.
When we multiply two numbers, the result, which is called the product, will have a sign that depends on the signs of the two numbers being multiplied together: 1. Positive × Positive = Positive 2. Negative × Negative = Positive 3. Positive × Negative = Negative 4. Negative × Positive = Negative The first two cases show us that when the signs are the same, the result is always positive.
However, when the signs are different, as in the third and fourth cases, the result is always negative, regardless of the magnitude of the numbers.
This means,Therefore, based on the rules of multiplication for positive and negative numbers, the conjecture that correctly describes the product of a positive number and a negative number is: b) Always negative This covers all cases where one number is positive and the other is negative. Hence, we can confidently say that whenever you multiply a positive number by a negative number, the product will be negative.