Question

Select the correct answer.

Let f(X) and g(x) be polynomials as shown below.
f(x) = a0+a1x+a2x^2...+anx^n
g(x)=b0+b1x+b2x^2...bmx^m

Which of the following is true about f(x) and g(x)?
A.
F(x) and g(x) are not closed under subtraction because when subtracted, the result will not be polynomial
B.
f(x) and g(x) are closed under subtraction because when subtracted, the result will be a polynomial.
C. f(x) and g(x) are closed under subtraction because when subtracted, the result will not be a polynomial.
D. f(x) and g(x) are not closed under subtraction because when subtracted, the result will be a polynomial.

Answer 1

B.

f(x) and g(x) are closed under subtraction because when subtracted, the result will be a polynomial.

Explanation:

`f(x)-g(x)\\=(a_0+a_1x+a_2x^2...+a_nx^n)-(b_0+b_1x+b_2x^2...b_mx^m)`

Assume m>=n WOLOG

`=(a_0-b_0)+(a_1x-b_1x)+(a_2x^2-b_2x^2)+...+(a_nx^n-b_nx^n)-b_((n+1))x^(n+1)-...-b_(m-1)x^(m-1)-b_mx^m`

`=(a_0-b_0)+(a_1-b_1)x+(a_2-b_2)x^2+...+(a_n-b_n)x^n-b_((n+1))x^(n+1)-...-b_(m-1)x^(m-1)-b_mx^m`

Are also a polynomial in general, but maybe in a different degree

Closure under subtraction means, for all polynomials, under the operation of subtraction, it also belongs to polynomials