Answer :
To determine which student is correct about the polynomial being in standard form, we need to understand what standard form means for polynomials. Generally, in standard form, the terms of the polynomial are arranged in descending order of their degrees.
### Step-by-Step Analysis
#### Marcus' Polynomial: [tex]\(3x^3 - 4x^2y + y^3 + 2\)[/tex]
List the terms and their degrees:
1. [tex]\(3x^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(3x^3\)[/tex] only involves [tex]\(x\)[/tex] to the power of [tex]\(3\)[/tex]).
2. [tex]\(-4x^2y\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(x^2y\)[/tex] involves [tex]\(x\)[/tex] to the power of [tex]\(2\)[/tex] and [tex]\(y\)[/tex] to the power of [tex]\(1\)[/tex], so [tex]\(2 + 1 = 3\)[/tex]).
3. [tex]\(y^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(y^3\)[/tex] only involves [tex]\(y\)[/tex] to the power of [tex]\(3\)[/tex]).
4. [tex]\(2\)[/tex]: Degree is [tex]\(0\)[/tex] (constant term).
Looking at the terms in Marcus' polynomial, we see they are ordered as follows:
- [tex]\(3x^3\)[/tex] (degree 3)
- [tex]\(-4x^2y\)[/tex] (degree 3)
- [tex]\(y^3\)[/tex] (degree 3)
- [tex]\(2\)[/tex] (degree 0)
#### Ariel's Polynomial: [tex]\(y^3 - 4x^2y + 3x^3 + 2\)[/tex]
List the terms and their degrees:
1. [tex]\(y^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(y^3\)[/tex] only involves [tex]\(y\)[/tex] to the power of [tex]\(3\)[/tex]).
2. [tex]\(-4x^2y\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(x^2y\)[/tex] involves [tex]\(x\)[/tex] to the power of [tex]\(2\)[/tex] and [tex]\(y\)[/tex] to the power of [tex]\(1\)[/tex], so [tex]\(2 + 1 = 3\)[/tex]).
3. [tex]\(3x^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(3x^3\)[/tex] only involves [tex]\(x\)[/tex] to the power of [tex]\(3\)[/tex]).
4. [tex]\(2\)[/tex]: Degree is [tex]\(0\)[/tex] (constant term).
Looking at the terms in Ariel's polynomial, we see they are ordered as follows:
- [tex]\(y^3\)[/tex] (degree 3)
- [tex]\(-4x^2y\)[/tex] (degree 3)
- [tex]\(3x^3\)[/tex] (degree 3)
- [tex]\(2\)[/tex] (degree 0)
### Conclusion
Comparing both polynomials:
- Both polynomials have terms of the same degree.
- Both Marcus' and Ariel's polynomials have their terms involving [tex]\(x\)[/tex] and/or [tex]\(y\)[/tex] of the same degree, but the order of the terms is different.
However, neither arrangement follows a clear pattern of descending order by the sum of the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Thus, neither Marcus nor Ariel is strictly correct according to the conventional understanding of the standard form of polynomials by degree.
For polynomial terms with the same total degree, typically terms would be ordered by descending exponents of one variable (usually [tex]\(x\)[/tex], and then by [tex]\(y\)[/tex] if the other variables do not change the total degree). However, since there’s ambiguity and neither student maintains a strict order with descending variable degrees in an established convention, thus:
- Neither Marcus nor Ariel is fully correct.
### Step-by-Step Analysis
#### Marcus' Polynomial: [tex]\(3x^3 - 4x^2y + y^3 + 2\)[/tex]
List the terms and their degrees:
1. [tex]\(3x^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(3x^3\)[/tex] only involves [tex]\(x\)[/tex] to the power of [tex]\(3\)[/tex]).
2. [tex]\(-4x^2y\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(x^2y\)[/tex] involves [tex]\(x\)[/tex] to the power of [tex]\(2\)[/tex] and [tex]\(y\)[/tex] to the power of [tex]\(1\)[/tex], so [tex]\(2 + 1 = 3\)[/tex]).
3. [tex]\(y^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(y^3\)[/tex] only involves [tex]\(y\)[/tex] to the power of [tex]\(3\)[/tex]).
4. [tex]\(2\)[/tex]: Degree is [tex]\(0\)[/tex] (constant term).
Looking at the terms in Marcus' polynomial, we see they are ordered as follows:
- [tex]\(3x^3\)[/tex] (degree 3)
- [tex]\(-4x^2y\)[/tex] (degree 3)
- [tex]\(y^3\)[/tex] (degree 3)
- [tex]\(2\)[/tex] (degree 0)
#### Ariel's Polynomial: [tex]\(y^3 - 4x^2y + 3x^3 + 2\)[/tex]
List the terms and their degrees:
1. [tex]\(y^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(y^3\)[/tex] only involves [tex]\(y\)[/tex] to the power of [tex]\(3\)[/tex]).
2. [tex]\(-4x^2y\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(x^2y\)[/tex] involves [tex]\(x\)[/tex] to the power of [tex]\(2\)[/tex] and [tex]\(y\)[/tex] to the power of [tex]\(1\)[/tex], so [tex]\(2 + 1 = 3\)[/tex]).
3. [tex]\(3x^3\)[/tex]: Degree is [tex]\(3\)[/tex] (since [tex]\(3x^3\)[/tex] only involves [tex]\(x\)[/tex] to the power of [tex]\(3\)[/tex]).
4. [tex]\(2\)[/tex]: Degree is [tex]\(0\)[/tex] (constant term).
Looking at the terms in Ariel's polynomial, we see they are ordered as follows:
- [tex]\(y^3\)[/tex] (degree 3)
- [tex]\(-4x^2y\)[/tex] (degree 3)
- [tex]\(3x^3\)[/tex] (degree 3)
- [tex]\(2\)[/tex] (degree 0)
### Conclusion
Comparing both polynomials:
- Both polynomials have terms of the same degree.
- Both Marcus' and Ariel's polynomials have their terms involving [tex]\(x\)[/tex] and/or [tex]\(y\)[/tex] of the same degree, but the order of the terms is different.
However, neither arrangement follows a clear pattern of descending order by the sum of the powers of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Thus, neither Marcus nor Ariel is strictly correct according to the conventional understanding of the standard form of polynomials by degree.
For polynomial terms with the same total degree, typically terms would be ordered by descending exponents of one variable (usually [tex]\(x\)[/tex], and then by [tex]\(y\)[/tex] if the other variables do not change the total degree). However, since there’s ambiguity and neither student maintains a strict order with descending variable degrees in an established convention, thus:
- Neither Marcus nor Ariel is fully correct.
