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

Given polynomial: [tex]\( 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 \)[/tex]

A. [tex]\(-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3\)[/tex]

B. [tex]\(-13y^4 + x^3y - 8x^2y^2 + 9xy^3 + 3x^4\)[/tex]

C. [tex]\(3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 13y^4\)[/tex]

D. [tex]\(3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4\)[/tex]



Answer :

To find the correct polynomial that combines like terms and expresses the given polynomial in standard form, we should combine the like terms step-by-step. Let's break down the expression:

Given polynomial:
[tex]\[ 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 \][/tex]

1. Combine the [tex]\( y^4 \)[/tex] terms:
[tex]\[ -4y^4 - 9y^4 = -13y^4 \][/tex]

2. Combine the [tex]\( x^2y^2 \)[/tex] terms:
[tex]\[ -10x^2y^2 + 2x^2y^2 = -8x^2y^2 \][/tex]

3. There are no other like terms to combine, so let's write the polynomial combining the results:
[tex]\[ 3x^4 + x^3y + 9xy^3 - 13y^4 - 8x^2y^2 \][/tex]

Rewriting in decreasing order of degrees, we get:
[tex]\[ 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \][/tex]

We now need to compare this with the given options to see which polynomial matches this form.

The options are:

1. [tex]\( 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 \)[/tex]
2. [tex]\( -13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3 \)[/tex]
3. [tex]\( -13y^4 + x^3y - 8x^2y^2 + 9xy^3 + 3x^4 \)[/tex]
4. [tex]\( 3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 13y^4 \)[/tex]
5. [tex]\( 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \)[/tex]

The polynomial that matches our combined expression is indeed:

[tex]\[ 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \][/tex]

Therefore, the corresponding option is:

[tex]\[ 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \][/tex]

Thus, the correct polynomial is [tex]\( 3x^4 + x^3y - 8x^2y^2 + 9xy^3 - 13y^4 \)[/tex]. This matches the fifth option.