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

Question

Accepted Answer

### 1. "An integer multiplied by an integer is an integer."
# Brief Explanations:
The set of integers is closed under multiplication. By the definition of integers (..., -2, -1, 0, 1, 2, ...) and the rules of multiplication, multiplying any two integers (positive, negative, or zero) results in another integer. For example, $2 \times 3 = 6$, $(-2) \times 3 = -6$, $0 \times 5 = 0$, all of which are integers. So this statement is always true.
# Answer: always

### 2. "An integer added to an integer is an integer."
# Brief Explanations:
Integers are closed under addition. When adding two integers (e.g., $3 + (-5) = -2$, $0 + 7 = 7$, $-4 + (-1) = -5$), the result is always an integer. This is a fundamental property of the integer set, so the statement is always true.
# Answer: always

### 3. "An integer divided by an integer is an integer."
# Brief Explanations:
Dividing integers does not always yield an integer. For example, $5 \div 2 = 2.5$, where 5 and 2 are integers, but the result is not an integer. However, there are cases where it is an integer (e.g., $6 \div 3 = 2$). So the result can sometimes be an integer and sometimes not.
# Answer: sometimes

### 4. "A polynomial added to a polynomial is a polynomial."
# Brief Explanations:
When adding two polynomials, we combine like terms. The sum of two polynomials will have terms with non - negative integer exponents (since the exponents in the sum will be the same as in the original polynomials, and the coefficients will be real numbers). For example, if we have $f(x)=x^{2}+2x + 1$ and $g(x)=3x^{2}-x + 5$, then $f(x)+g(x)=(1 + 3)x^{2}+(2-1)x+(1 + 5)=4x^{2}+x + 6$, which is also a polynomial. So the set of polynomials is closed under addition, and the statement is always true.
# Answer: always

### 5. "A polynomial divided by a polynomial is a polynomial."
# Brief Explanations:
Dividing polynomials does not always result in a polynomial. For example, if we divide the polynomial $x + 1$ by the polynomial $x$, we get $1+\frac{1}{x}$, which is not a polynomial (because of the term $\frac{1}{x}=x^{-1}$ with a negative exponent). However, there are cases where the division results in a polynomial, like dividing $x^{2}+2x + 1$ by $x + 1$ (result is $x + 1$). So the statement is sometimes true.
# Answer: sometimes

### 6. "A polynomial subtracted from a polynomial is a polynomial."
# Brief Explanations:
Subtracting two polynomials is equivalent to adding the first polynomial and the negative of the second polynomial. Since polynomials are closed under addition, and the negative of a polynomial is also a polynomial (multiplying a polynomial by - 1), the result of subtracting two polynomials will be a polynomial. For example, if $f(x)=2x^{3}+x$ and $g(x)=x^{3}-2x$, then $f(x)-g(x)=(2x^{3}+x)-(x^{3}-2x)=2x^{3}+x - x^{3}+2x=x^{3}+3x$, which is a polynomial. So the statement is always true.
# Answer: always

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Question

Explanation:

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

Answer:

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