Let's go through the process of combining like terms and putting the polynomial in standard form step-by-step:
1. List all the given polynomial terms:
[tex]\[
-5 x^3 y^3, \quad 8 x^4 y^2, \quad -x y^5, \quad -2 x^2 y^4, \quad 8 x^6, \quad 3 x^2 y^4, \quad -6 x y^5
\][/tex]
2. Combine the like terms:
- For [tex]\(x^6\)[/tex], there is only one term:
[tex]\[
8 x^6
\][/tex]
- For [tex]\(x^4 y^2\)[/tex], there is only one term:
[tex]\[
8 x^4 y^2
\][/tex]
- For [tex]\(x^3 y^3\)[/tex], there is only one term:
[tex]\[
-5 x^3 y^3
\][/tex]
- For [tex]\(x y^5\)[/tex], combine the terms [tex]\(-x y^5\)[/tex] and [tex]\(-6 x y^5\)[/tex]:
[tex]\[
-x y^5 - 6 x y^5 = -7 x y^5
\][/tex]
- For [tex]\(x^2 y^4\)[/tex], combine the terms [tex]\(-2 x^2 y^4\)[/tex] and [tex]\(3 x^2 y^4\)[/tex]:
[tex]\[
-2 x^2 y^4 + 3 x^2 y^4 = x^2 y^4
\][/tex]
3. Write the polynomial in standard form by arranging the terms in descending order based on the degree of [tex]\(x\)[/tex]:
The polynomial in standard form is:
[tex]\[
8 x^6 + 8 x^4 y^2 - 5 x^3 y^3 + x^2 y^4 - 7 x y^5
\][/tex]
Therefore, the polynomial that combines like terms and is written in standard form is:
[tex]\[
8 x^6 + 8 x^4 y^2 - 5 x^3 y^3 + x^2 y^4 - 7 x y^5
\][/tex]
This matches the fourth option:
[tex]\[
8 x^6 + 8 x^4 y^2 - 5 x^3 y^3 + x^2 y^4 - 7 x y^5
\][/tex]
