Which is true about the completely simplified difference of the polynomials 6*x*6 − *x*3*y*4 − 5*xy*5 and 4*x*5*y* + 2*x*3*y*4 + 5*xy*5?

The difference has 3 terms and a degree of 6.

The difference has 4 terms and a degree of 6.

The difference has 3 terms and a degree of 7.

The difference has 4 terms and a degree of 7.

To solve this problem, we need to subtract the second polynomial from the first and then simplify the result.

Given polynomials:
1. \( 6x^6 - x^3y^4 - 5xy^5 \)
2. \( 4x^5y + 2x^3y^4 + 5xy^5 \)

### Step 1: Subtract the polynomials
\[
(6x^6 - x^3y^4 - 5xy^5) - (4x^5y + 2x^3y^4 + 5xy^5)
\]

Distribute the negative sign:
\[
6x^6 - x^3y^4 - 5xy^5 - 4x^5y - 2x^3y^4 - 5xy^5
\]

### Step 2: Combine like terms
- \( 6x^6 \) has no like terms, so it stays as \( 6x^6 \).
- Combine the \( x^3y^4 \) terms: \( -x^3y^4 - 2x^3y^4 = -3x^3y^4 \)
- Combine the \( xy^5 \) terms: \( -5xy^5 - 5xy^5 = -10xy^5 \)
- The \( 4x^5y \) term has no like terms, so it stays as \( -4x^5y \).

So, the simplified difference is:
\[
6x^6 - 4x^5y - 3x^3y^4 - 10xy^5
\]

### Step 3: Determine the number of terms and the degree
- There are **4 terms** in the simplified expression.
- The highest degree term is \( 6x^6 \), which has a degree of **6**.

### Conclusion
The correct statement is:
- **The difference has 4 terms and a degree of 6.**

Let me know if you'd like more details or if you have any questions.

Here are 5 related questions:
1. What is the degree of a polynomial, and how is it determined?
2. How do you identify like terms in a polynomial expression?
3. Can you explain how to subtract polynomials step by step?
4. How does the degree of a polynomial affect its graph?
5. What are the properties of polynomials that make them suitable for algebraic operations?

**Tip:** Always align like terms vertically when adding or subtracting polynomials to avoid mistakes.

Learn how to simplify the difference of two polynomials: 6x^6 - x^3y^4 - 5xy^5 and 4x^5y + 2x^3y^4 + 5xy^5. This problem involves subtracting polynomials and combining like terms. Suitable for Grades 9-12.

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