for each statement, choose whether it is always, sometimes, or never true. an integer subtracted from an…

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

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

Answer

Explanation:

Step1: Analyze integer subtraction

The set of integers is closed under subtraction. If (a) and (b) are integers, (a - b) is also an integer. So, an integer subtracted from an integer is always an integer.

Step2: Analyze integer multiplication

The set of integers is closed under multiplication. If (a) and (b) are integers, (a\times b) is an integer. So, an integer multiplied by an integer is always an integer.

Step3: Analyze integer division

If (a = 4) and (b = 2), (\frac{a}{b}=2) (an integer), but if (a = 5) and (b = 2), (\frac{a}{b}=2.5) (not an integer). So, an integer divided by an integer is sometimes an integer.

Step4: Analyze polynomial addition

When adding two polynomials, we combine like - terms. The result of adding two polynomials (P(x)) and (Q(x)) is (P(x)+Q(x)), which is still a polynomial. So, a polynomial added to a polynomial is always a polynomial.

Step5: Analyze polynomial division

If (P(x)=x^{2}-1) and (Q(x)=x - 1), (\frac{P(x)}{Q(x)}=x + 1) (a polynomial), but if (P(x)=x) and (Q(x)=x^{2}), (\frac{P(x)}{Q(x)}=\frac{1}{x}) (not a polynomial). So, a polynomial divided by a polynomial is sometimes a polynomial.

Step6: Analyze polynomial subtraction

When subtracting two polynomials (P(x)) and (Q(x)) to get (P(x)-Q(x)), we combine like - terms. The result is still a polynomial. So, a polynomial subtracted from a polynomial is always a polynomial.

Answer:

An integer subtracted from an integer is an integer: Always An integer multiplied by an integer is an integer: Always An integer divided by an integer is an integer: Sometimes A polynomial added to a polynomial is a polynomial: Always A polynomial divided by a polynomial is a polynomial: Sometimes A polynomial subtracted from a polynomial is a polynomial: Always