Which polynomial correctly combines the like terms and puts the given polynomial in standard form?

Given polynomial:
[tex]-5x^3y^3 + 8x^4y^2 - xy^5 - 2x^2y^4 + 8x^6 + 3x^2y^4 - 6xy^5[/tex]

A. [tex]-7xy^5 + 5x^2y^4 - 5x^3y^3 + 8x^4y^2 + 8x^6[/tex]

B. [tex]5xy^5 + 8x^4y^2 + x^2y^4 - 5x^3y^3 + 8x^6[/tex]

C. [tex]8x^6 + 5xy^5 + 8x^4y^2 + x^2y^4 - 5x^3y^3[/tex]

D. [tex]8x^6 + 8x^4y^2 - 5x^3y^3 + x^2y^4 - 7xy^5[/tex]



Answer :

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]