Let's start by analyzing the given polynomial:
[tex]\[ 2x^2y + 8x^3 - xy^2 - 2x^3 + 3xy^2 + 6y^3 \][/tex]
We need to combine like terms to simplify the polynomial. Like terms are terms that have the same variables raised to the same powers.
First, let's identify and combine all the [tex]\( x^3 \)[/tex] terms:
[tex]\[ 8x^3 - 2x^3 = 6x^3 \][/tex]
Next, let's combine the [tex]\( x^2y \)[/tex] terms:
[tex]\[ 2x^2y \][/tex]
(No other similar terms are present, so this term remains alone.)
Now, let's combine the [tex]\( xy^2 \)[/tex] terms:
[tex]\[ xy^2 + 3xy^2 = 2xy^2 + 3xy^2 = 2xy^2 + 5xy^2 = 5xy^2 \][/tex]
Finally, the [tex]\( y^3 \)[/tex] term on its own:
[tex]\[ 6y^3 \][/tex]
(No other similar terms are present, so this term remains alone.)
Putting all the combined terms together, we have:
[tex]\[ 6x^3 + 2x^2y + 2xy^2 + 6y^3 \][/tex]
So after simplifying, the polynomial is:
[tex]\[ 6x^3 + 2x^2y + 5xy^2 + 6y^3 \][/tex]
We need to write this polynomial in standard form, ordered by the degree of each term, starting from the highest degree. The highest degree term in this polynomial is [tex]\( 6x^3 \)[/tex] which has a degree of 3.
Hence, the first term is:
[tex]\[ 6x^3 \][/tex]
So, the first term of the simplified polynomial is:
[tex]\[ 6x^3 \][/tex]
The correct answer is:
[tex]\[ \boxed{6x^3} \][/tex]
