Determine if the expression \( 9 c^{5}+5 c^{3} \) is a polynomial or not. If it is a polynomial, state the type and degree of the polynomial. Answer Attempt 1 out of 2 The given expression a polynomial.

Yes, the expression \( 9c^{5} + 5c^{3} \) is a **polynomial**. Let's break down why this is the case and identify its type and degree. ### **Definition of a Polynomial** A **polynomial** is an algebraic expression composed of terms that are: - **Variables** raised to **non-negative integer exponents**. - **Coefficients**, which are real numbers multiplying the variables. - Combined using **addition** or **subtraction**. ### **Analyzing the Given Expression** Let's examine each term in the expression \( 9c^{5} + 5c^{3} \): 1. **First Term: \( 9c^{5} \)** - **Coefficient:** 9 - **Variable:** \( c \) - **Exponent:** 5 (a non-negative integer) 2. **Second Term: \( 5c^{3} \)** - **Coefficient:** 5 - **Variable:** \( c \) - **Exponent:** 3 (a non-negative integer) Both terms satisfy the criteria for being part of a polynomial: - The exponents of \( c \) are non-negative integers. - The variables are combined using addition. ### **Type of the Polynomial** - **Univariate Polynomial:** Since there's only one variable (\( c \)) involved, it's a univariate polynomial. ### **Degree of the Polynomial** - The **degree** of a polynomial is the highest power of the variable present in the expression. - In \( 9c^{5} + 5c^{3} \), the highest exponent is **5**. ### **Conclusion** - **Polynomial:** Yes, \( 9c^{5} + 5c^{3} \) is a polynomial. - **Type:** Univariate polynomial. - **Degree:** 5. This detailed analysis confirms that the given expression is indeed a polynomial of degree five in the variable \( c \).