Answer: An integer added to an integer is an integer, this statement is always true. A polynomial subtracted from a polynomial is a polynomial, this statement is always true. A polynomial divided by a polynomial is a polynomial, this statement is sometimes true. A polynomial multiplied by a polynomial is a polynomial, this statement is always true.
Explanation:
1)
The closure property of integer states that the addition, subtraction and multiplication is integers is always an integer.
If [tex]a\in Z\text{ and }b\in Z[/tex], then a+b\in Z.
Therefore, an integer added to an integer is an integer, this statement is always true.
2)
A polynomial is in the form of,
[tex]p(x)=a_nx^n+a_{n-1}x^{x-1}+...+a_1x+a_0[/tex]
Where [tex]a_n,a_{n-1},...,a_1,a_0[/tex] are constant coefficient.
When we subtract the two polynomial then the resultant is also a polynomial form.
Therefore, a polynomial subtracted from a polynomial is a polynomial, this statement is always true.
3)
If a polynomial divided by a polynomial then it may or may not be a polynomial.
If the degree of numerator polynomial is higher than the degree of denominator polynomial then it may be a polynomial.
For example:
[tex]f(x)=x^2-2x+5x-10 \text{ and } g(x)=x-2[/tex]
Then [tex]\frac{f(x)}{g(x)}=x^2+5[/tex], which a polynomial.
If the degree of numerator polynomial is less than the degree of denominator polynomial then it is a rational function.
For example:
[tex]f(x)=x^2-2x+5x-10 \text{ and } g(x)=x-2[/tex]
Then [tex]\frac{g(x)}{f(x)}=\frac{1}{x^2+5}[/tex], which a not a polynomial.
Therefore, a polynomial divided by a polynomial is a polynomial, this statement is sometimes true.
4)
As we know a polynomial is in the form of,
[tex]p(x)=a_nx^n+a_{n-1}x^{x-1}+...+a_1x+a_0[/tex]
Where [tex]a_n,a_{n-1},...,a_1,a_0[/tex] are constant coefficient.
When we multiply the two polynomial, the degree of the resultand function is addition of degree of both polyminals and the resultant is also a polynomial form.
Therefore, a polynomial subtracted from a polynomial is a polynomial, this statement is always true.
1) an integer added to an integer is an integer - Always
(adding two integers always results in an integer, if the two integers are positive, their sum will be positive, if two integers are negative, they will yield a negative sum)
2) a polynomial subtracted from a polynomial is a polynomial - Always
(if the polynomials are subtracted vertically, then the signs of the subtracted polynomial's need to be flipped to their opposites)
3) a polynomial divided by a polynomial is a polynomial - Sometimes
(this depends on the degree of polynomials in the numerator and the denominator)
4) a polynomial multiplied by a polynomial is a polynomial - Always
(when two polynomials are added, the product function is the addition of degree to both the polynomials)
