To determine why [tex]\( f(x) = x^3 + 3|x| - 5 \)[/tex] is not a polynomial function, we need to understand the definition of a polynomial function.
A polynomial function is a mathematical expression involving a sum of powers of the variable [tex]\( x \)[/tex], with non-negative integer exponents, multiplied by coefficients which are real numbers. The general form of a polynomial function is:
[tex]\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \][/tex]
where [tex]\( a_n, a_{n-1}, \ldots, a_1, a_0 \)[/tex] are constants (real numbers), and [tex]\( n \)[/tex] is a non-negative integer.
Now, let's analyze the given function [tex]\( f(x) = x^3 + 3|x| - 5 \)[/tex]:
- The term [tex]\( x^3 \)[/tex] is a polynomial term because it follows the form [tex]\( x^n \)[/tex] with [tex]\( n \)[/tex] being a non-negative integer (in this case, [tex]\( n = 3 \)[/tex]).
- The term [tex]\( -5 \)[/tex] is also a polynomial term, because constants are considered valid polynomial terms with [tex]\( n = 0 \)[/tex] (i.e., [tex]\( x^0 = 1 \)[/tex]).
- The term [tex]\( 3|x| \)[/tex] involves the absolute value of [tex]\( x \)[/tex], which is not of the form [tex]\( x^n \)[/tex] for any non-negative integer [tex]\( n \)[/tex].
The presence of the absolute value [tex]\( |x| \)[/tex] makes the function [tex]\( f(x) \)[/tex] not a polynomial function because the absolute value term does not fit into the standard definition of a polynomial term. Polynomial functions do not include absolute values, square roots, or other non-polynomial operations on [tex]\( x \)[/tex].
Therefore, the correct answer is:
C. The function is not a polynomial function because of the presence of [tex]\( |x| \)[/tex].
