Let's analyze and simplify the polynomial expression given:
[tex]\[ -3x^3 - x^2y - 3xy^2 + y^3 \][/tex]
To break this down step-by-step:
1. Understand the Polynomial Structure:
This polynomial consists of four different terms.
- The first term is [tex]\(-3x^3\)[/tex].
- The second term is [tex]\(-x^2y\)[/tex].
- The third term is [tex]\(-3xy^2\)[/tex].
- The fourth term is [tex]\(y^3\)[/tex].
2. Identify the Degree of Each Term:
- The degree of the term [tex]\(-3x^3\)[/tex] is 3.
- The degree of the term [tex]\(-x^2y\)[/tex] is [tex]\(2+1 = 3\)[/tex].
- The degree of the term [tex]\(-3xy^2\)[/tex] is [tex]\(1+2 = 3\)[/tex].
- The degree of the term [tex]\(y^3\)[/tex] is 3.
All terms are of degree 3, meaning the given polynomial is a homogeneous polynomial of degree 3.
3. Combine Like Terms:
There are no like terms to combine in this expression, as all terms are distinct.
4. Factorization (if possible):
We can check if it's possible to factor the polynomial. However, in this case, a quick inspection suggests that the polynomial doesn't factor nicely into simpler polynomials with integer coefficients.
5. Verification of the Polynomial:
Let's verify the constructed polynomial to ensure that there are no errors:
[tex]\[ -3x^3 - x^2y - 3xy^2 + y^3 \][/tex]
Hence, the polynomial expression given is correct and does not simplify further. We've analyzed it thoroughly and confirmed the structure:
[tex]\[ \boxed{-3x^3 - x^2y - 3xy^2 + y^3} \][/tex]
