X^3y + x^2 y^2- xy^3 is a given algebraic expression . Why is the given expression a polynomial? 

The given expression is a polynomial because it satisfies the following conditions that define polynomials:

1. **Variables with non-negative integer exponents**: In the expression \(x^3y + x^2y^2 - xy^3\), all the variables \(x\) and \(y\) are raised to non-negative integer exponents (3, 2, 1, etc.). This is a key requirement for polynomials.

2. **Finite number of terms**: The expression consists of three distinct terms: \(x^3y\), \(x^2y^2\), and \(-xy^3\). Each term is a product of constants and variables raised to non-negative integer exponents.

3. **No variables in denominators or under radicals**: Polynomials cannot have variables in the denominator or under radicals (such as square roots), and this expression adheres to that rule.

4. **Addition or subtraction of terms**: The terms are combined using addition and subtraction, which is permissible for polynomials.

Thus, the given expression is a polynomial.

Would you like further clarification or more details on any part of this explanation?

Here are five related questions to expand your understanding:
1. What are the degrees of the terms in the polynomial \(x^3y + x^2y^2 - xy^3\)?
2. How do you find the degree of a polynomial with multiple variables?
3. Can a polynomial contain negative exponents?
4. What is the degree of the entire polynomial \(x^3y + x^2y^2 - xy^3\)?
5. How can you factor a polynomial like the one given in the problem?

**Tip:** The degree of a polynomial is determined by the term with the highest total degree, which is the sum of the exponents of the variables in that term.

X^3y + x^2 y^2 - xy^3 is a given algebraic expression. Why is the given expression a polynomial?

Understand why the algebraic expression X^3y + x^2 y^2 - xy^3 is a polynomial. This explanation covers the properties of polynomials, including non-negative integer exponents, the structure of terms, and the absence of variables in denominators or radicals. Perfect for Grades 8-10 students.

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