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For each statement, choose whether it is always, sometimes, or never true.
An integer multiplied by an integer is an integer.
An integer divided by an integer is an integer.
An integer subtracted from 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.

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Answer with explanation:

A)

  • An integer multiplied by an integer is an integer.

This option is always true.

Since, the integer is closed under multiplication.

B)

  • An integer divided by an integer is an integer.

This option is not always true. It is sometimes true.

Because:

when 4 is divided by 2 then we get:

4÷2=2 which is an integer

But

3÷5 is not an integer it is a rational number.

C)

An integer subtracted from an integer is an integer.

This option is always true.

Since, the set of integers is closed under subtraction.

D)

A polynomial added to a polynomial is a polynomial.

This option is always true.

Because the set of polynomials is closed under addition.

E)

A polynomial divided by a polynomial is a polynomial.

This option is not always true.It is sometimes true.

Since if we divide x³ by x then we get the result as:

x³÷x=x² which is a polynomial.

but when we divide x by x³ then we get the result as:

x÷x³=1/x² which is not a polynomial function.

It is a rational function.

F)

A polynomial subtracted from a polynomial is a polynomial.

This is always true.

Because the set of polynomial is closed under subtraction.

An integer multiplied by an integer is an integer. (Always true)

Reason: The set of integers is closed under multiplication

An integer divided by an integer is an integer. (Sometimes true)

Reason: A set of integers is not closed under division. However, it is sometimes possible to get an integer as a result of the division of two integers. For example, 4/2  =  2

An integer subtracted from an integer is an integer. (Always true)

Reason: A set of integers is closed under subtraction

A polynomial added to a polynomial is a polynomial. (Always true)

Reason: Addition of two polynomials always gives another polynomial as a result of grouping similar elements together

A polynomial divided by a polynomial is a polynomial. (Sometimes true)

Reason: Division of two polynomials sometimes gives a polynomial as the result, but not all the time

A polynomial subtracted from a polynomial is a polynomial. (Always true)

Reason: Subtraction of two polynomials always gives another polynomial as a result of grouping similar elements together

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