Five negative integers and four positive integers are being multiplied. Find the sign of the product.

Options

A. negative integer
B. positive integer
C. cannot be determined
D. none of the above

Answer

To determine the sign of the product when multiplying five negative integers and four positive integers, we can follow these steps:

### Step 1: Identify the number of negative and positive integers
- We have **5 negative integers** and **4 positive integers**.

### Step 2: Understand the rules of multiplication for signs
- The rules for multiplication of integers are:
- A negative integer multiplied by another negative integer results in a **positive** integer.
- A negative integer multiplied by a positive integer results in a **negative** integer.
- A positive integer multiplied by a positive integer results in a **positive** integer.

### Step 3: Determine the overall sign from the counts
- Since we have **5 negative integers**:
- An even number of negative integers multiplied together results in a **positive** product.
- An odd number of negative integers multiplied together results in a **negative** product.

- Here, we have **5 negative integers**, which is an **odd** number.

### Step 4: Combine the signs with positive integers
- We also have **4 positive integers**:
- The product of any number of positive integers is always **positive**.

### Step 5: Calculate the final sign
- Since we have an **odd number of negative integers (5)**, the product of these negative integers will be **negative**.
- The positive integers do not change the fact that the overall product remains **negative** because multiplying a negative number by a positive number results in a negative number.

### Conclusion
- Therefore, the sign of the product of five negative integers and four positive integers is **negative**.

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