Certainly! Let's break down and identify each term and their respective coefficients in the polynomial expression [tex]\( 3x^2 y^2 - xy z + z^3 \)[/tex]:
1. First Term: [tex]\( 3x^2 y^2 \)[/tex]
- Coefficient: [tex]\( 3 \)[/tex]
- Degree of [tex]\( x \)[/tex]: [tex]\( 2 \)[/tex]
- Degree of [tex]\( y \)[/tex]: [tex]\( 2 \)[/tex]
- Degree of [tex]\( z \)[/tex]: [tex]\( 0 \)[/tex]
- Representation: [tex]\((2, 2, 0)\)[/tex]
2. Second Term: [tex]\(-xyz\)[/tex]
- Coefficient: [tex]\( -1 \)[/tex] (since there is an implied 1 with the negative sign)
- Degree of [tex]\( x \)[/tex]: [tex]\( 1 \)[/tex]
- Degree of [tex]\( y \)[/tex]: [tex]\( 1 \)[/tex]
- Degree of [tex]\( z \)[/tex]: [tex]\( 1 \)[/tex]
- Representation: [tex]\((1, 1, 1)\)[/tex]
3. Third Term: [tex]\( z^3 \)[/tex]
- Coefficient: [tex]\( 1 \)[/tex] (since there is an implied positive 1)
- Degree of [tex]\( x \)[/tex]: [tex]\( 0 \)[/tex] (since [tex]\( x \)[/tex] is not present in this term)
- Degree of [tex]\( y \)[/tex]: [tex]\( 0 \)[/tex] (since [tex]\( y \)[/tex] is not present in this term)
- Degree of [tex]\( z \)[/tex]: [tex]\( 3 \)[/tex]
- Representation: [tex]\((0, 0, 3)\)[/tex]
Summarizing the terms and their coefficients:
- [tex]\( 3x^2 y^2 \)[/tex] corresponds to [tex]\(((2, 2, 0), 3)\)[/tex]
- [tex]\(-xyz\)[/tex] corresponds to [tex]\(((1, 1, 1), -1)\)[/tex]
- [tex]\( z^3 \)[/tex] corresponds to [tex]\(((0, 0, 3), 1)\)[/tex]
Thus, the polynomial [tex]\( 3x^2 y^2 - xy z + z^3 \)[/tex] consists of the terms and their coefficients as:
```
[((2, 2, 0), 3), ((1, 1, 1), -1), ((0, 0, 3), 1)]
```
