In the proof that ab+cd , where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd is a rational number.(1 point) Responses By the Closure Property, ad+bc and bd are both integers, and so ad+bcbd is a quotient of two integers. By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers. By the Closure Property, a quotient of imaginary numbers is a rational number. By the Closure Property, a quotient of imaginary numbers is a rational number. By the Closure Property, ad+bcbd reduces to a whole number, which is a subset of rational numbers. By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers. By the Closure Property, ad+bc and bd are both quotients of integers, and so ad+bcbd is a quotient of two integers.

The correct explanation for why \( \frac{ad + bc}{bd} \) is a rational number involves understanding the properties of integers and rational numbers.

To clarify, let's analyze the expression \( \frac{ad + bc}{bd} \):

1. **Identifying the Parts**:
- In this case, \( ad + bc \) is a sum of products of integers. Since \( a, b, c, \) and \( d \) are all integers, both \( ad \) and \( bc \) are also integers. Therefore, their sum \( ad + bc \) is also an integer.

2. **Denominator**:
- The product \( bd \) is also an integer because it is the product of two integers (\( b \) and \( d \)), and since both \( b \) and \( d \) are non-zero, \( bd \) is non-zero.

3. **Quotients of Integers**:
- A rational number is defined as any number that can be expressed as a quotient \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). In this case, \( ad + bc \) is an integer, and \( bd \) is a non-zero integer. Thus, \( \frac{ad + bc}{bd} \) represents a quotient of two integers where the denominator is non-zero.

4. **Conclusion**:
- Therefore, since both the numerator (\( ad + bc \)) and denominator (\( bd \)) are integers with \( bd \neq 0\), the expression \( \frac{ad + bc}{bd} \) is indeed a rational number.

The most accurate response from your options is the second one:

"By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so the fraction with numerator \( ad + bc \) and denominator \( bd \) is a quotient of two integers."

This captures the essence of why the result is a rational number.