https://file.aiquickdraw.com/mathbot/file/1726321260431jicloqub.jpg

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.

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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.

Find the degree of each of the following polynomials:
1) 7x^2 y^3 z^4 - 5x^4 y^2 z^2 + x^4 yz
2) 2 - 3x^2 + 4x^4 - 5x^6
3) y^3 - 7y^2 + 5y + 1
4) 3x^2 y^3 - x^3 y^2 + 2x^4 y + 4x^5
5) z^3 - 10x^5

Learn how to find the degree of multivariable polynomials with step-by-step solutions. This problem includes various algebraic expressions with multiple variables. Suitable for students in Grades 9-12.

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