Let's find the degree of each polynomial listed.

General Rule:

• The degree of a polynomial is the highest sum of the exponents of the variables in any single term.

1. $7x^2 y^3 z^4 - 5x^4 y^2 z^2 + x^4 yz$

• For the first term: $7x^2 y^3 z^4$, the sum of the exponents is $2 + 3 + 4 = 9$.
• For the second term: $5x^4 y^2 z^2$, the sum of the exponents is $4 + 2 + 2 = 8$.
• For the third term: $x^4 yz$, the sum of the exponents is $4 + 1 + 1 = 6$.

Degree: The highest sum is 9, so the degree of this polynomial is 9.

2. $2 - 3x^2 + 4x^4 - 5x^6$

• For the constant $2$, the degree is 0.
• For the term $3x^2$, the degree is 2.
• For the term $4x^4$, the degree is 4.
• For the term $5x^6$, the degree is 6.

Degree: The highest degree is 6, so the degree of this polynomial is 6.

3. $y^3 - 7y^2 + 5y + 1$

• For the term $y^3$, the degree is 3.
• For the term $7y^2$, the degree is 2.
• For the term $5y$, the degree is 1.
• The constant $1$ has degree 0.

Degree: The highest degree is 3, so the degree of this polynomial is 3.

4. $3x^2 y^3 - x^3 y^2 + 2x^4 y + 4x^5$

• For the term $3x^2 y^3$, the sum of the exponents is $2 + 3 = 5$.
• For the term $x^3 y^2$, the sum of the exponents is $3 + 2 = 5$.
• For the term $2x^4 y$, the sum of the exponents is $4 + 1 = 5$.
• For the term $4x^5$, the degree is 5.

Degree: The highest sum is 5, so the degree of this polynomial is 5.

5. $z^3 - 10x^5$

• For the term $z^3$, the degree is 3.
• For the term $10x^5$, the degree is 5.

Degree: The highest degree is 5, so the degree of this polynomial is 5.

Final Answers:

1. Degree = 9
2. Degree = 6
3. Degree = 3
4. Degree = 5
5. Degree = 5

Let me know if you would like further clarification or details on any part of the process.

Here are 5 relative questions to expand your understanding:

1. How do you determine the degree of a single-variable polynomial?
2. What is the degree of a constant term, and why?
3. Can you explain why the degree of a polynomial depends on the sum of exponents in multivariable cases?
4. How does the degree of a polynomial affect its graph?
5. What are some examples of real-world applications where knowing the degree of a polynomial is important?

Tip: In a multivariable polynomial, always check all terms, even if they involve different combinations of variables, to ensure you're identifying the highest sum of exponents for the degree.