Sure, let's assign the correct options to each statement based on the true answers we have available:
1. The sum of two nonzero rational numbers is rational.
- Always True: When you add two rational numbers, the result is always another rational number. Rational numbers are closed under addition.
2. The sum of a nonzero rational number and an irrational number is rational.
- Never True: Adding a nonzero rational number to an irrational number always results in an irrational number. The rational number cannot "cancel out" the irrational nature of the other number.
3. The product of two irrational numbers is irrational.
- Sometimes True: While multiplying two irrational numbers often gives an irrational result, there are exceptions. For example, the product of [tex]\(\sqrt{2}\)[/tex] and [tex]\(\sqrt{2}\)[/tex] is 2, which is rational.
4. The product of a nonzero rational number and an irrational number is rational.
- Never True: Multiplying a nonzero rational number by an irrational number always results in an irrational number. The rational number cannot alter the irrational nature of the result.
Let’s fill in the table accordingly:
\begin{tabular}{|c|c|}
\hline
Statement & \begin{tabular}{c}
Always, \\
Sometimes, or \\
Never True
\end{tabular} \\
\hline
The sum of two nonzero rational numbers is rational. & Always True \\
\hline
The sum of a nonzero rational number and an irrational number is rational. & Never True \\
\hline
The product of two irrational numbers is irrational. & Sometimes True \\
\hline
The product of a nonzero rational number and an irrational number is rational. & Never True \\
\hline
\end{tabular}
