To solve this problem, let's carefully analyze each statement regarding the polynomial [tex]\( p(x) \)[/tex] and the integer [tex]\( b \)[/tex].
Given: [tex]\( p(x) \)[/tex] is a polynomial, and [tex]\( b \)[/tex] is an integer.
A polynomial [tex]\( p(x) \)[/tex] can be expressed in general form as:
[tex]\[ p(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n \][/tex]
where [tex]\( a_i \)[/tex] are coefficients which could be integers, real numbers, etc.
Now let's analyze the operations:
1. Sum of [tex]\( p(x) \)[/tex] and [tex]\( b \)[/tex]:
Consider the sum of [tex]\( p(x) \)[/tex] and [tex]\( b \)[/tex], denoted as [tex]\( p(x) + b \)[/tex].
[tex]\[ p(x) + b = (a_0 + b) + a_1 x + a_2 x^2 + \ldots + a_n x^n \][/tex]
Since adding a constant (integer [tex]\( b \)[/tex]) to the constant term [tex]\( a_0 \)[/tex] in the polynomial does not change the nature of the terms involving [tex]\( x \)[/tex], the result is still a polynomial of the same degree.
2. Difference between [tex]\( p(x) \)[/tex] and [tex]\( b \)[/tex]:
Consider the difference [tex]\( p(x) - b \)[/tex].
[tex]\[ p(x) - b = (a_0 - b) + a_1 x + a_2 x^2 + \ldots + a_n x^n \][/tex]
Similarly, subtracting an integer [tex]\( b \)[/tex] from the constant term [tex]\( a_0 \)[/tex] does not change the nature of the polynomial. The result is still a polynomial.
From these analyses, we can conclude that the sum and difference of a polynomial [tex]\( p(x) \)[/tex] and an integer [tex]\( b \)[/tex] are both polynomials. Therefore, the correct answer is:
The sum and difference of [tex]\( p(x) \)[/tex] and [tex]\( b \)[/tex] are polynomials.
