Certainly! To explore and simplify the expression given, [tex]\(2x^2 - 5xy - y^3\)[/tex], we will highlight the components and structure of the expression step by step.
1. Identify the Terms:
The expression consists of three distinct terms:
[tex]\[
2x^2, \; -5xy, \; \text{and} \; -y^3
\][/tex]
2. First Term: [tex]\(2x^2\)[/tex]
- This term is a quadratic term involving [tex]\(x\)[/tex].
- It means [tex]\(2\)[/tex] times [tex]\(x\)[/tex] raised to the power of [tex]\(2\)[/tex].
- Its degree with respect to [tex]\(x\)[/tex] is [tex]\(2\)[/tex] and with respect to [tex]\(y\)[/tex] is [tex]\(0\)[/tex].
3. Second Term: [tex]\(-5xy\)[/tex]
- This term is a product of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] multiplied by [tex]\(-5\)[/tex].
- It is a first degree term in both [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- Its degree with respect to [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is each [tex]\(1\)[/tex].
4. Third Term: [tex]\(-y^3\)[/tex]
- This term involves [tex]\(y\)[/tex] cubed and is multiplied by [tex]\(-1\)[/tex].
- It is a cubic term in [tex]\(y\)[/tex].
- Its degree with respect to [tex]\(x\)[/tex] is [tex]\(0\)[/tex] and with respect to [tex]\(y\)[/tex] is [tex]\(3\)[/tex].
5. Combine the Terms:
- We now write the expression bringing together all these identified parts:
[tex]\[
2x^2 - 5xy - y^3
\][/tex]
6. Structure Analysis:
- The expression is a polynomial in two variables, [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- Each term has specific degrees of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- There appear to be no like terms that can be combined any further, thus the expression is already in its simplest form.
In conclusion, the expression [tex]\(2x^2 - 5xy - y^3\)[/tex] is a properly simplified polynomial composed of a quadratic term in [tex]\(x\)[/tex], a linear term in both [tex]\(x\)[/tex] and [tex]\(y\)[/tex], and a cubic term in [tex]\(y\)[/tex].
