To determine which of the given statements is true about the polynomials [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex], let's analyze the subtraction of these polynomials.
Given:
[tex]\[ f(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n \][/tex]
[tex]\[ g(x) = b_0 + b_1 x + b_2 x^2 + \ldots + b_m x^m \][/tex]
When we subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex], the result is as follows:
[tex]\[ f(x) - g(x) = (a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n) - (b_0 + b_1 x + b_2 x^2 + \ldots + b_m x^m) \][/tex]
This can be simplified to:
[tex]\[ f(x) - g(x) = (a_0 - b_0) + (a_1 - b_1)x + (a_2 - b_2)x^2 + \ldots + (a_n - b_n)x^n \][/tex]
Here, each coefficient of the resulting polynomial is the difference of the corresponding coefficients of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Since the result of subtracting two polynomials is another polynomial (where the term with the higher degree among [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] could remain, and new coefficients are simply the differences of the original coefficients), this demonstrates that polynomials are closed under subtraction. Closure under an operation means that performing the operation on members of a set results in another member of that set.
Therefore, the correct statement is:
D. [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are closed under subtraction because when subtracted, the result will be a polynomial.
