To classify the polynomial $5x + 3x^4 - 7x^3 + 10$, we need to analyze the degree and the number of terms.

Step 1: Degree of the Polynomial

The degree of a polynomial is the highest power of $x$ in the expression. In this case: $5x \text{ (degree 1)}, \quad 3x^4 \text{ (degree 4)}, \quad -7x^3 \text{ (degree 3)}, \quad 10 \text{ (degree 0)}.$ The highest degree is 4, so this is a fourth-degree polynomial.

Step 2: Number of Terms

The number of terms is the count of distinct terms in the polynomial. The terms are: $5x, \quad 3x^4, \quad -7x^3, \quad 10.$ This gives 4 terms, so the polynomial is a quartic polynomial (fourth degree) with 4 terms.

Final Answer:

Fourth degree polynomial

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Would you like a breakdown of how to classify polynomials or more details about degrees and terms?

Related Questions:

1. How do you determine the degree of a polynomial with multiple variables?
2. What is the difference between monomials, binomials, and polynomials?
3. Can a polynomial have a negative degree?
4. How do you classify a first-degree polynomial?
5. How do constants like $10$ affect the classification of a polynomial?

Tip: Always identify the highest power of the variable for the degree and count distinct terms for the classification!