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

Question

Accepted Answer

### 1. An integer added to an integer is an integer.
# Brief Explanations:
Integers are closed under addition. For any two integers $a$ and $b$, $a + b$ is also an integer. For example, $2+3 = 5$ (both integers, sum is integer), $-1+4=3$ (both integers, sum is integer). So this statement is always true.
# Answer: always true

### 2. An integer subtracted from an integer is an integer.
# Brief Explanations:
Integers are closed under subtraction. If we have two integers $m$ and $n$, $m - n$ is an integer. For example, $5-3 = 2$, $-2 - (-1)=-1$. So this statement is always true.
# Answer: always true

### 3. An integer multiplied by an integer is an integer.
# Brief Explanations:
Integers are closed under multiplication. For any integers $x$ and $y$, $x\times y$ is an integer. For example, $3\times4 = 12$, $-2\times5=-10$. So this statement is always true.
# Answer: always true

### 4. A polynomial divided by a polynomial is a polynomial.
# Brief Explanations:
Consider an example: Let the dividend be $x + 1$ (a polynomial) and the divisor be $x$ (a polynomial). Then $\frac{x + 1}{x}=1+\frac{1}{x}$, which is not a polynomial (since $\frac{1}{x}=x^{- 1}$ has a negative exponent, and polynomials have non - negative integer exponents). However, if we divide $x^{2}+2x$ by $x$ (where $x
eq0$), we get $x + 2$ (a polynomial). So sometimes the division of two polynomials gives a polynomial, and sometimes it does not.
# Answer: sometimes true

### 5. A polynomial subtracted from a polynomial is a polynomial.
# Brief Explanations:
Polynomials are closed under subtraction. Let $P(x)=a_{n}x^{n}+\cdots+a_{1}x + a_{0}$ and $Q(x)=b_{m}x^{m}+\cdots+b_{1}x + b_{0}$ be two polynomials. When we subtract them ($P(x)-Q(x)$ or $Q(x)-P(x)$), the result will still be a sum of terms with non - negative integer exponents and coefficients (after combining like terms). For example, $(x^{2}+2x + 1)-(x + 1)=x^{2}+x$, which is a polynomial. So this statement is always true.
# Answer: always true

### 6. A polynomial multiplied by a polynomial is a polynomial.
# Brief Explanations:
When we multiply two polynomials, we use the distributive property (FOIL for binomials, and extended for polynomials of higher degrees). The product of two terms with non - negative integer exponents will result in a term with a non - negative integer exponent (by adding exponents when multiplying like bases, e.g., $x^{2}\times x^{3}=x^{5}$). Combining these terms will still give a polynomial. For example, $(x + 1)(x - 1)=x^{2}-1$, which is a polynomial. So this statement is always true.
# Answer: always true

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QUESTION IMAGE

Question

Explanation:

1. An integer added to an integer is an integer.

Answer:

2. An integer subtracted from an integer is an integer.