Question

for each statement, choose whether it is always, sometimes, or never true.
an integer divided by an integer is an integer. select
an integer added to an integer is an integer. select
an integer subtracted from an integer is an integer. select
a polynomial multiplied by a polynomial is a polynomial. select
a polynomial added to a polynomial is a polynomial. select
a polynomial subtracted from a polynomial is a polynomial. select

Explanation:

1. "An integer divided by an integer is an integer"

Step1: Analyze integer division

Consider examples: \( 4 \div 2 = 2 \) (integer), but \( 4 \div 3=\frac{4}{3} \) (not an integer). So it's not always true, just sometimes.

2. "An integer added to an integer is an integer"

Step1: Recall integer addition property

By the closure property of integers under addition, for any integers \( a,b \), \( a + b \) is an integer. So this is always true.

3. "An integer subtracted from an integer is an integer"

Step1: Recall integer subtraction property

For any integers \( a,b \), \( a - b \) is an integer (since subtraction is addition of the opposite). So this is always true.

4. "A polynomial multiplied by a polynomial is a polynomial"

Answer:

s:

• An integer divided by an integer is an integer: sometimes
• An integer added to an integer is an integer: always
• An integer subtracted from an integer is an integer: always
• A polynomial multiplied by a polynomial is a polynomial: always
• A polynomial added to a polynomial is a polynomial: always
• A polynomial subtracted from a polynomial is a polynomial: always