Determine if the expression minus, a, plus, 3, a, to the power 5 , minus, a, squared−a+3a
5
−a
2
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.

The expression given is:

\[
-a + 3a^5 - a^2
\]

### Step 1: Check if the expression is a polynomial
A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers, and the operations involved are addition, subtraction, and multiplication.

In this case, the expression involves the following terms:
- \(-a\) (which is \(a^1\))
- \(3a^5\) 
- \(-a^2\)

All the terms involve the variable \(a\) raised to non-negative integer exponents (1, 5, and 2). Therefore, the expression **is** a polynomial.

### Step 2: Determine the type and degree of the polynomial
- **Type**: The type of a polynomial is determined by the number of terms it has. The given expression has three terms, so it is a **trinomial**.
  
- **Degree**: The degree of a polynomial is the highest exponent of the variable in the expression. Here, the highest exponent is 5 (from the term \(3a^5\)). Therefore, the degree of the polynomial is **5**.

### Final Answer:
The expression is a polynomial. It is a **trinomial** of degree **5**.

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Would you like more details or have any questions? Here are some related questions you might consider:

1. What is the general form of a polynomial expression?
2. How can you determine the degree of a polynomial with multiple variables?
3. What are the different types of polynomials based on the number of terms?
4. How does the leading coefficient affect the graph of a polynomial?
5. What is the significance of the degree of a polynomial in determining its end behavior?

**Tip:** Always check the exponents of all terms when identifying a polynomial to ensure they are non-negative integers.

Understand and analyze the polynomial expression -a + 3a^5 - a^2. Learn how to determine if an expression is a polynomial, identify its type (trinomial), and calculate its degree (5). Suitable for Grades 10-12.

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