To combine like terms and put the given polynomial in standard form, follow these steps:
1. List the Given Polynomial Terms:
[tex]\[
-5x^3y^3 + 8x^4y^2 - x y^5 - 2 x^2 y^4 + 8 x^6 + 3 x^2 y^4 - 6 x y^5
\][/tex]
2. Combine Like Terms:
- Combine terms with [tex]\(x^3y^3\)[/tex]: [tex]\(-5 x^3 y^3\)[/tex]
- Combine terms with [tex]\(x^4y^2\)[/tex]: [tex]\(8 x^4 y^2\)[/tex]
- Combine terms with [tex]\(xy^5\)[/tex]: [tex]\(- x y^5\)[/tex] and [tex]\(- 6 x y^5\)[/tex] gives:
[tex]\[
(- x y^5) + (- 6 x y^5) = - 7 x y^5
\][/tex]
- Combine terms with [tex]\(x^2y^4\)[/tex]: [tex]\(-2 x^2 y^4\)[/tex] and [tex]\(3 x^2 y^4\)[/tex] gives:
[tex]\[
(-2 x^2 y^4) + (3 x^2 y^4) = 1 x^2 y^4
\][/tex]
- Combine terms with [tex]\(x^6\)[/tex]: [tex]\(8 x^6\)[/tex]
3. Form the Standard Form Polynomial: Order the terms by decreasing total degree (sum of the exponents), and if the total degrees are the same, then order by the exponent of [tex]\(x\)[/tex].
The combined terms are:
[tex]\[
8 x^6, \quad 8 x^4 y^2, \quad - 5 x^3 y^3, \quad 1 x^2 y^4, \quad - 7 x y^5
\][/tex]
4. Write the Polynomial in Standard Form:
[tex]\[
8 x^6 + 8 x^4 y^2 - 5 x^3 y^3 + 1 x^2 y^4 - 7 x y^5
\][/tex]
Thus, the polynomial that correctly combines the like terms and puts the given 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 correct answer is:
[tex]\[
8 x^6 + 8 x^4 y^2 - 5 x^3 y^3 + x^2 y^4 - 7 x y^5
\][/tex]
