find the sum or difference. then identify the degree of the sum or difference and classify the polynomial by the number of terms. for this equation. (3 + 7x^3 + x^4) - (8 - x + x^4)

Let's break down and solve the given polynomial expression:

### Step 1: Distribute the negative sign and combine like terms
The given expression is:

\[
(3 + 7x^3 + x^4) - (8 - x + x^4)
\]

Now distribute the negative sign to each term in the second polynomial:

\[
3 + 7x^3 + x^4 - 8 + x - x^4
\]

### Step 2: Combine like terms
1. Combine the constant terms:
   \[
   3 - 8 = -5
   \]

2. Combine the \(x\) terms:
   \[
   +x \quad (\text{there's only one }x\text{-term})
   \]

3. Combine the \(x^3\) terms:
   \[
   7x^3 \quad (\text{there's only one }x^3\text{-term})
   \]

4. Combine the \(x^4\) terms:
   \[
   x^4 - x^4 = 0
   \]

Thus, the resulting polynomial is:

\[
-5 + x + 7x^3
\]

### Step 3: Identify the degree of the polynomial
The degree of a polynomial is the highest power of \(x\) in the expression. In this case, the highest power of \(x\) is \(x^3\), so the degree is:

\[
\text{Degree} = 3
\]

### Step 4: Classify by the number of terms
This polynomial has three terms: \(-5\), \(x\), and \(7x^3\). A polynomial with three terms is called a **trinomial**.

### Final Answer:
- **Sum/Difference**: \(-5 + x + 7x^3\)
- **Degree**: 3
- **Classification**: Trinomial

Would you like more details or further explanation?

Here are 5 related questions for further exploration:
1. How do you handle subtraction of polynomials with negative exponents?
2. What is the degree of a polynomial with several variables (e.g., \(x^2y^3\))?
3. Can a polynomial have more than one highest-degree term?
4. How do you classify a polynomial with only one term?
5. What is the role of the leading coefficient in determining the behavior of a polynomial?

**Tip:** Always double-check that you've combined like terms, especially when subtracting polynomials with the same variable powers.

Learn how to find the sum or difference of polynomials with a step-by-step solution. Identify the degree and classify the polynomial by the number of terms. Explore algebraic concepts such as combining like terms and polynomial classification. Suitable for high school students (Grades 9-12).

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