To determine whether the function [tex]\( f(x) = x + \pi \)[/tex] is a polynomial, we need to understand what constitutes a polynomial.
A polynomial is an algebraic expression that consists of terms involving variables raised to non-negative integer exponents, multiplied by coefficients, and combined using addition, subtraction, and multiplication. The general form of a polynomial in one variable [tex]\( x \)[/tex] is:
[tex]\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants (coefficients), and [tex]\( n \)[/tex] is a non-negative integer representing the degree of the polynomial.
Now, let's analyze the given function [tex]\( f(x) = x + \pi \)[/tex]:
1. Variable and Exponents: In the expression [tex]\( x + \pi \)[/tex], the term [tex]\( x \)[/tex] is our variable, which can be seen as [tex]\( x^1 \)[/tex]. The exponent [tex]\( 1 \)[/tex] is a non-negative integer, satisfying the criteria for polynomials.
2. Coefficients: The term [tex]\(\pi\)[/tex], although it is an irrational number, is a constant. In polynomials, constants are allowed and treated just as coefficients.
3. Operations: The operations involved in [tex]\( x + \pi \)[/tex] are addition, which is valid for polynomial expressions.
Given the above points, [tex]\( f(x) = x + \pi \)[/tex] meets all the criteria for being a polynomial:
- It involves the variable [tex]\( x \)[/tex] with a non-negative integer exponent.
- It includes a constant term [tex]\(\pi\)[/tex], which is permissible.
- The operations used are appropriate for polynomials.
Therefore, the function [tex]\( f(x) = x + \pi \)[/tex] is indeed a polynomial.
The correct answer is:
A. True
