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Determine whether the following function is a polynomial function. If the function is a polynomial, state its degree. If it is not, explain why. Write the polynomial in standard form. Then identify the leading term and the constant term.

[tex]\[ F(x) = 9x^6 - \pi x^5 + \frac{1}{2} \][/tex]

Determine whether [tex]\( F(x) \)[/tex] is a polynomial or not. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. It is not a polynomial because the variable [tex]\( x \)[/tex] is raised to the [tex]\(\square\)[/tex] power, which is not a nonnegative integer.
(Type an integer or a fraction.)

B. It is a polynomial of degree [tex]\(\square\)[/tex].
(Type an integer or a fraction.)

C. It is not a polynomial because the function is the ratio of two distinct polynomials, and the polynomial in the denominator is of positive degree.



Answer :

GinnyAnswer
To determine whether the function [tex]\( F(x) = 9x^6 - \pi x^5 + \frac{1}{2} \)[/tex] is a polynomial, we need to check if the function satisfies the requirements of a polynomial

### Definition of a Polynomial

1. A polynomial is a sum of terms where each term is a product of a constant coefficient and a nonnegative integer power of the variable [tex]\( x \)[/tex].
2. The powers of [tex]\( x \)[/tex] must be nonnegative integers (i.e., whole numbers including zero).
3. The coefficients can be any real number.

### Checking the Function [tex]\( F(x) \)[/tex]

The given function is:
[tex]\[ F(x) = 9x^6 - \pi x^5 + \frac{1}{2} \][/tex]

#### Terms Analysis

1. First Term: [tex]\( 9x^6 \)[/tex]
- The coefficient is 9, which is a real number.
- The exponent of [tex]\( x \)[/tex] is 6, a nonnegative integer.

2. Second Term: [tex]\( -\pi x^5 \)[/tex]
- The coefficient is [tex]\( -\pi \)[/tex], which is a real number (since [tex]\( \pi \)[/tex] is a constant real number).
- The exponent of [tex]\( x \)[/tex] is 5, a nonnegative integer.

3. Third Term: [tex]\( \frac{1}{2} \)[/tex]
- This can be seen as [tex]\( \frac{1}{2} \times x^0 \)[/tex] where the exponent of [tex]\( x \)[/tex] is 0, a nonnegative integer.
- The coefficient is [tex]\( \frac{1}{2} \)[/tex], which is a real number.

Since all the terms have nonnegative integer exponents and real coefficients, we can conclude that [tex]\( F(x) \)[/tex] is a polynomial function.

### Standard Form, Degree, Leading Term, and Constant Term

- Standard Form:
The polynomial is already written in standard form as [tex]\( F(x) = 9x^6 - \pi x^5 + \frac{1}{2} \)[/tex], where the terms are arranged in descending order of their exponents.

- Degree:
The degree of a polynomial is the highest power of the variable [tex]\( x \)[/tex]. In this case, the highest power of [tex]\( x \)[/tex] is 6.

- Leading Term:
The leading term is the term with the highest power of [tex]\( x \)[/tex]. Here, the leading term is [tex]\( 9x^6 \)[/tex].

- Constant Term:
The constant term is the term without any variable [tex]\( x \)[/tex]. In this polynomial, the constant term is [tex]\( \frac{1}{2} \)[/tex].

### Conclusion

The correct assessment of the function is:

Choice B:
It is a polynomial of degree 6.

Therefore, filling in the blanks:

B. It is a polynomial of degree [tex]\( \mathbf{6} \)[/tex].

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