To determine whether the function [tex]\( f(x) = 6x^4 + 7x^3 \)[/tex] is a polynomial function, we need to check if it meets the criteria for a polynomial. Let's go through each step systematically:
1. Definition of a Polynomial:
A polynomial function is a mathematical expression that can be represented in the form:
[tex]\[
P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0
\][/tex]
where [tex]\( n \)[/tex] is a non-negative integer, and [tex]\( a_n, a_{n-1}, \ldots, a_0 \)[/tex] are constants.
2. Form of the Given Function:
The given function is:
[tex]\[
f(x) = 6x^4 + 7x^3
\][/tex]
Here, we can see that the terms have powers of [tex]\( x \)[/tex] that are non-negative integers (4 and 3), and the coefficients (6 and 7) are constants.
3. Check the Polynomial Criteria:
- Each term in the function [tex]\( 6x^4 \)[/tex] and [tex]\( 7x^3 \)[/tex] has a power of [tex]\( x \)[/tex] that is a non-negative integer.
- The coefficients for each term are constants.
Since both criteria are met, [tex]\( f(x) = 6x^4 + 7x^3 \)[/tex] is indeed a polynomial function.
4. Identify the Degree:
The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex] in the expression. In this function, the highest power of [tex]\( x \)[/tex] is 4 (from the term [tex]\( 6x^4 \)[/tex]).
Therefore, we conclude that:
- The function [tex]\( f(x) = 6x^4 + 7x^3 \)[/tex] is a polynomial.
- The degree of the polynomial is [tex]\( 4 \)[/tex].
Hence, the correct choice is:
A. It is a polynomial. The degree of the polynomial is [tex]\( 4 \)[/tex].
