Classify the following on the basis of order, number of variables, and number of terms:

x4 + y5 + 3x2y + 7y - 4x3y3 + 2xyz - 3yx2 - 4

Let's classify the given expression \( x^4 + y^5 + 3x^2y + 7y - 4x^3y^3 + 2xyz - 3yx^2 - 4 \) based on the order, number of variables, and number of terms.

### 1. Number of Terms:
A term in an algebraic expression is a part separated by plus (+) or minus (−) signs. The given expression consists of the following terms:
1. \( x^4 \)
2. \( y^5 \)
3. \( 3x^2y \)
4. \( 7y \)
5. \( -4x^3y^3 \)
6. \( 2xyz \)
7. \( -3yx^2 \)
8. \( -4 \)

So, **there are 8 terms** in the expression.

### 2. Number of Variables:
The variables in the expression are the different letters used to represent the unknowns. The expression contains three variables:
- \( x \)
- \( y \)
- \( z \)

So, **there are 3 variables** in the expression.

### 3. Order of the Expression:
The order of the expression is determined by the highest degree of any term in the expression. The degree of a term is the sum of the exponents of the variables in that term.

Let's calculate the degree of each term:
1. \( x^4 \): Degree \( = 4 \)
2. \( y^5 \): Degree \( = 5 \)
3. \( 3x^2y \): Degree \( = 2 + 1 = 3 \)
4. \( 7y \): Degree \( = 1 \)
5. \( -4x^3y^3 \): Degree \( = 3 + 3 = 6 \)
6. \( 2xyz \): Degree \( = 1 + 1 + 1 = 3 \)
7. \( -3yx^2 \): Degree \( = 2 + 1 = 3 \)
8. \( -4 \): Degree \( = 0 \) (constant term)

The highest degree among these is 6 (from the term \( -4x^3y^3 \)). 

So, **the order of the expression is 6**.

### Summary:
- **Number of terms:** 8
- **Number of variables:** 3
- **Order:** 6

Would you like more details or have any questions?

### Related Questions:
1. How do you calculate the degree of a polynomial with multiple variables?
2. What are the types of algebraic expressions based on the number of terms?
3. How do you classify a polynomial by degree and number of terms?
4. How can you identify like terms in a polynomial?
5. What is the difference between the degree of a term and the degree of a polynomial?

**Tip:** When working with polynomials, always check each term's degree by summing up the exponents of the variables in that term.

Learn how to classify algebraic expressions based on their order, number of variables, and number of terms. This detailed explanation covers identifying terms, counting variables, and determining the highest degree in the expression. Suitable for high school students studying algebra.

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