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.