To simplify and rearrange the polynomial [tex]\( 2x^2 y + 8x^3 - xy^2 - 2x^3 + 3xy^2 + 6y^3 \)[/tex] into its standard form, we need to collect the like terms and combine them step-by-step.
Let’s break this down:
1. Identify Like Terms:
- [tex]\( 8x^3 \)[/tex] and [tex]\( -2x^3 \)[/tex]
- [tex]\( 2x^2y \)[/tex]
- [tex]\( -xy^2 \)[/tex] and [tex]\( 3xy^2 \)[/tex]
- [tex]\( 6y^3 \)[/tex]
2. Combine Like Terms:
- For the [tex]\( x^3 \)[/tex] terms:
[tex]\[
8x^3 - 2x^3 = 6x^3
\][/tex]
- For the [tex]\( x^2y \)[/tex] terms:
[tex]\[
2x^2y
\][/tex]
- For the [tex]\( xy^2 \)[/tex] terms:
[tex]\[
-xy^2 + 3xy^2 = 2xy^2
\][/tex]
- The [tex]\( y^3 \)[/tex] term remains:
[tex]\[
6y^3
\][/tex]
3. Construct the Simplified Polynomial:
Combining all the simplified like terms, we get:
[tex]\[
6x^3 + 2x^2y + 2xy^2 + 6y^3
\][/tex]
4. Arrange in Standard Form (Decreasing Degree):
The polynomial is already arranged in descending order by the degree of each term:
[tex]\[
6x^3 + 2x^2y + 2xy^2 + 6y^3
\][/tex]
Given the simplified polynomial, let’s verify the first and final terms:
- First Term: [tex]\( 6x^3 \)[/tex]
- Final Term: As Raj noted, the final term is [tex]\( 6y^3 \)[/tex].
Therefore, the first term of the polynomial that Raj ended up with is:
[tex]\[
\boxed{6x^3}
\][/tex]
