To determine the simplified difference of the given polynomials and its properties, we need to follow a structured method to solve the problem. Let's analyze this step-by-step:
1. Identify the given polynomials:
[tex]\[
\text{poly1} = 6x^6 - x^3y^4 - 5xy^5
\][/tex]
[tex]\[
\text{poly2} = 4x^5y + 2x^3y^4 + 5xy^5
\][/tex]
2. Subtract the second polynomial from the first:
[tex]\[
\text{difference} = \text{poly1} - \text{poly2}
\][/tex]
Substituting the polynomials:
[tex]\[
\text{difference} = (6x^6 - x^3y^4 - 5xy^5) - (4x^5y + 2x^3y^4 + 5xy^5)
\][/tex]
3. Distribute the subtraction across the terms:
[tex]\[
\text{difference} = 6x^6 - x^3y^4 - 5xy^5 - 4x^5y - 2x^3y^4 - 5xy^5
\][/tex]
4. Combine like terms:
[tex]\[
\text{difference} = 6x^6 - (x^3y^4 + 2x^3y^4) - (5xy^5 + 5xy^5) - 4x^5y
\][/tex]
Simplifying the combined terms:
[tex]\[
\text{difference} = 6x^6 - 3x^3y^4 - 10xy^5 - 4x^5y
\][/tex]
5. Simplify the subtraction:
Here the simplified polynomial now reads:
[tex]\[
\text{difference} = 6x^6 - 4x^5y - 3x^3y^4 - 10xy^5
\][/tex]
6. Determine the degree of the simplified polynomial:
The degree of a polynomial is the highest degree of any single term. Examining the terms:
- [tex]\(6x^6\)[/tex]: degree 6
- [tex]\(-4x^5y\)[/tex]: degree 6 (since [tex]\(5 + 1 = 6\)[/tex])
- [tex]\(-3x^3y^4\)[/tex]: degree 7 (since [tex]\(3 + 4 = 7\)[/tex])
- [tex]\(-10xy^5\)[/tex]: degree 6 (since [tex]\(1 + 5 = 6\)[/tex])
The highest degree here is 7.
7. Count the number of terms:
The simplified polynomial [tex]\(6x^6 - 4x^5y - 3x^3y^4 - 10xy^5\)[/tex] has 4 distinct terms.
Based on these steps, the simplified difference has 4 terms and a degree of 7. Therefore, the true statement is:
The difference has 4 terms and a degree of 7.
