in the proof that ab+cdab+cd , where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbdad+bcbd is a rational number.(1 point) Responses 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+bcbdad+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+bcad+bc and bdbd are both integers, and so ad+bcbdad+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, ad+bcad+bc and bdbd are both quotients of integers, and so ad+bcbdad+bcbd is a quotient of two integers.

Answer 1

To explain why \( \frac{ad + bc}{bd} \) is a rational number, we can use the properties of integers and the definition of rational numbers.

First, by definition, a rational number is any number that can be expressed as the quotient of two integers, where the denominator is not zero.

In your case:

• \( a, b, c, \) and \( d \) are integers.
• Both \( b \) and \( d \) are non-zero.

Now, analyze the expression:

1. Numerator: The expression in the numerator is \( ad + bc \).

   • Since \( a, b, c, \) and \( d \) are all integers, the product \( ad \) is an integer (product of two integers is an integer).
   • Similarly, the product \( bc \) is also an integer.
   • Therefore, the sum \( ad + bc \) is an integer (sum of two integers is an integer).

2. Denominator: The denominator is \( bd \).

   • Since both \( b \) and \( d \) are non-zero integers, their product \( bd \) is also a non-zero integer.

Now, we have a fraction: \[ \frac{ad + bc}{bd} \]

• The numerator \( ad + bc \) is an integer, and the denominator \( bd \) is a non-zero integer.
• According to the definition of rational numbers, since we have an integer in the numerator and a non-zero integer in the denominator, \( \frac{ad + bc}{bd} \) is a rational number.

Therefore, the correct response to the explanation is "By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers." This is consistent with the properties of integers and their behavior under addition and multiplication.

Thus, \( \frac{ad + bc}{bd} \) is indeed a rational number.