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The degree of the polynomial  is 3. What is the value of m?

To find the value of \( m \) for which the degree of the polynomial is 3, let's analyze the polynomial given in the image:

\[
10x^2 + 2x^m y - 4y
\]

### Degree of the Polynomial
The degree of a polynomial is determined by the highest sum of the exponents in any single term.

1. **First term:** \( 10x^2 \) has a degree of \( 2 \) (since \( 2 + 0 = 2 \)).
2. **Second term:** \( 2x^m y \) has a degree of \( m + 1 \) (since the exponent of \( x \) is \( m \) and the exponent of \( y \) is \( 1 \)).
3. **Third term:** \( -4y \) has a degree of \( 1 \) (since the exponent of \( y \) is \( 1 \)).

Since the degree of the polynomial is given as \( 3 \), the term with the highest degree must equal 3. The highest degree is from the term \( 2x^m y \), which is \( m + 1 \).

### Equation:
\[
m + 1 = 3
\]

### Solving for \( m \):
\[
m = 3 - 1 = 2
\]

Thus, the value of \( m \) is \( \boxed{2} \).

Would you like further details or have any questions?

### Related Questions:
1. What happens to the degree if \( m \) is greater than 2?
2. How does the presence of \( y \) affect the degree of each term?
3. What is the degree of the polynomial if \( m \) is less than 2?
4. How do coefficients affect the degree of a polynomial?
5. Can you find a value of \( m \) that would make the polynomial's degree equal to 4?

### Tip:
Always remember that the degree of a polynomial is determined by the term with the highest sum of exponents, not just the highest exponent of a single variable.

Discover how to determine the value of m for a polynomial of degree 3. This problem involves analyzing polynomial terms and understanding how the degree is calculated. Suitable for high school students.

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