Answer :
To identify which polynomial is in standard form, we need to review the definitions and characteristics of a polynomial in standard form. A polynomial in standard form is expressed with its terms in descending order according to the degrees of each term.
Here are the given polynomials rewritten with an attention to their degrees:
1. [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex]:
- Degrees of the terms: [tex]\( xy \)[/tex] (degree 2), [tex]\( x^3y^2 \)[/tex] (degree 5), [tex]\( x^4y^3 \)[/tex] (degree 7), [tex]\( x^7y^4 \)[/tex] (degree 11).
- Descending order by degrees: [tex]\( 19x^7y^4 \)[/tex] (degree 11), [tex]\( -4x^4y^3 \)[/tex] (degree 7), [tex]\( 6x^3y^2 \)[/tex] (degree 5), [tex]\( 3xy \)[/tex] (degree 2).
- This polynomial is in standard form.
2. [tex]\( 18x^5 - 7x^2y - 2xy^2 + 17y^4 \)[/tex]:
- Degrees of the terms: [tex]\( x^5 \)[/tex] (degree 5), [tex]\( x^2y \)[/tex] (degree 3), [tex]\( xy^2 \)[/tex] (degree 3), [tex]\( y^4 \)[/tex] (degree 4).
- Descending order by degrees: [tex]\( 18x^5 \)[/tex] (degree 5), [tex]\( 17y^4 \)[/tex] (degree 4), [tex]\( -7x^2y \)[/tex] (degree 3), [tex]\( -2xy^2 \)[/tex] (degree 3).
- This polynomial is not in standard form because [tex]\( 17y^4 \)[/tex] comes after [tex]\( 18x^5 \)[/tex].
3. [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex]:
- Degrees of the terms: [tex]\( x^5y^5 \)[/tex] (degree 10), [tex]\( xy \)[/tex] (degree 2), [tex]\( x^2y^2 \)[/tex] (degree 4), constant term [tex]\( 12 \)[/tex] (degree 0).
- Descending order by degrees: [tex]\( x^5y^5 \)[/tex] (degree 10), [tex]\( -11x^2y^2 \)[/tex] (degree 4), [tex]\( -3xy \)[/tex] (degree 2), [tex]\( 12 \)[/tex] (degree 0).
- This polynomial is in standard form.
4. [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex]:
- Degrees of the terms: constant term [tex]\( 15 \)[/tex] (degree 0), [tex]\( xy^2 \)[/tex] (degree 3), [tex]\( x^9y^5 \)[/tex] (degree 14), [tex]\( x^7y^2 \)[/tex] (degree 9).
- Descending order by degrees: [tex]\( -11x^9y^5 \)[/tex] (degree 14), [tex]\( 5x^7y^2 \)[/tex] (degree 9), [tex]\( 12xy^2 \)[/tex] (degree 3), [tex]\( 15 \)[/tex] (degree 0).
- This polynomial is in standard form.
By comparing the polynomials and based on the degrees' arrangement:
1. [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex] - correctly in standard form (descending degrees).
2. [tex]\( 18x^5 - 7x^2y - 2xy^2 + 17y^4 \)[/tex] - order problems, not in standard form.
3. [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex] - correctly in standard form.
4. [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex] - correctly in standard form.
Thus, the correct answer is:
- [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex] (standard form)
- [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex] (standard form)
- [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex] (standard form)
But the polynomial that is verified as having the correct arrangement (degree-wise descending) and accurate adherence to the standard form structure is:
[tex]\[ 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \][/tex]
- Therefore, out of given options, the first polynomial is confidently the correct standard form polynomial based on our evaluations.
Here are the given polynomials rewritten with an attention to their degrees:
1. [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex]:
- Degrees of the terms: [tex]\( xy \)[/tex] (degree 2), [tex]\( x^3y^2 \)[/tex] (degree 5), [tex]\( x^4y^3 \)[/tex] (degree 7), [tex]\( x^7y^4 \)[/tex] (degree 11).
- Descending order by degrees: [tex]\( 19x^7y^4 \)[/tex] (degree 11), [tex]\( -4x^4y^3 \)[/tex] (degree 7), [tex]\( 6x^3y^2 \)[/tex] (degree 5), [tex]\( 3xy \)[/tex] (degree 2).
- This polynomial is in standard form.
2. [tex]\( 18x^5 - 7x^2y - 2xy^2 + 17y^4 \)[/tex]:
- Degrees of the terms: [tex]\( x^5 \)[/tex] (degree 5), [tex]\( x^2y \)[/tex] (degree 3), [tex]\( xy^2 \)[/tex] (degree 3), [tex]\( y^4 \)[/tex] (degree 4).
- Descending order by degrees: [tex]\( 18x^5 \)[/tex] (degree 5), [tex]\( 17y^4 \)[/tex] (degree 4), [tex]\( -7x^2y \)[/tex] (degree 3), [tex]\( -2xy^2 \)[/tex] (degree 3).
- This polynomial is not in standard form because [tex]\( 17y^4 \)[/tex] comes after [tex]\( 18x^5 \)[/tex].
3. [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex]:
- Degrees of the terms: [tex]\( x^5y^5 \)[/tex] (degree 10), [tex]\( xy \)[/tex] (degree 2), [tex]\( x^2y^2 \)[/tex] (degree 4), constant term [tex]\( 12 \)[/tex] (degree 0).
- Descending order by degrees: [tex]\( x^5y^5 \)[/tex] (degree 10), [tex]\( -11x^2y^2 \)[/tex] (degree 4), [tex]\( -3xy \)[/tex] (degree 2), [tex]\( 12 \)[/tex] (degree 0).
- This polynomial is in standard form.
4. [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex]:
- Degrees of the terms: constant term [tex]\( 15 \)[/tex] (degree 0), [tex]\( xy^2 \)[/tex] (degree 3), [tex]\( x^9y^5 \)[/tex] (degree 14), [tex]\( x^7y^2 \)[/tex] (degree 9).
- Descending order by degrees: [tex]\( -11x^9y^5 \)[/tex] (degree 14), [tex]\( 5x^7y^2 \)[/tex] (degree 9), [tex]\( 12xy^2 \)[/tex] (degree 3), [tex]\( 15 \)[/tex] (degree 0).
- This polynomial is in standard form.
By comparing the polynomials and based on the degrees' arrangement:
1. [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex] - correctly in standard form (descending degrees).
2. [tex]\( 18x^5 - 7x^2y - 2xy^2 + 17y^4 \)[/tex] - order problems, not in standard form.
3. [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex] - correctly in standard form.
4. [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex] - correctly in standard form.
Thus, the correct answer is:
- [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex] (standard form)
- [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex] (standard form)
- [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex] (standard form)
But the polynomial that is verified as having the correct arrangement (degree-wise descending) and accurate adherence to the standard form structure is:
[tex]\[ 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \][/tex]
- Therefore, out of given options, the first polynomial is confidently the correct standard form polynomial based on our evaluations.
