determine whether the following function is a polynomial function. if the function is a polynomial function, state its degree. if it is not, tell why not. write the polynomial in standard form. then identify the leading term and the constant term.

\f(x) = 5x^7 - \pi x^2 + \frac{1}{6}\

Question

determine whether the following function is a polynomial function. if the function is a polynomial function, state its degree. if it is not, tell why not. write the polynomial in standard form. then identify the leading term and the constant term.

\\f(x) = 5x^7 - \pi x^2 + \frac{1}{6}\\

\f(x) = 5x^7 - \pi x^2 + \frac{1}{6}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [], "current_concepts": [ "Polynomial Functions", "Degree and Leading Term", "Polynomial Standard Form" ] } ## Analyze the function definition $$ F(x) = 5x^7 - \pi x^2 + \frac{1}{6} $$ ## Determine if it is a polynomial $$ \begin{aligned} &\text{Exponents of } x: 7, 2, 0 \in \mathbb{Z}_{\ge 0}\ &\text{Coefficients: } 5, -\pi, \frac{1}{6} \in \mathbb{R}\ &\text{Therefore, } F(x) \text{ is a polynomial function.} \end{aligned} $$ ## Identify standard form, degree, leading term, and constant term $$ \begin{aligned} &\text{Standard Form: } F(x) = 5x^7 - \pi x^2 + \frac{1}{6}\ &\text{Degree: } 7\ &\text{Leading Term: } 5x^7\ &\text{Constant Term: } \frac{1}{6} \end{aligned} $$ The function is a polynomial function. - **Standard Form:** $F(x) = 5x^7 - \pi x^2 + \frac{1}{6}$ - **Degree:** $7$ - **Leading Term:** $5x^7$ - **Constant Term:** $\frac{1}{6}$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Functions" ] }

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Explanation:

Analyze the function definition

Determine if it is a polynomial

Identify standard form, degree, leading term, and constant term

Answer: