To determine which polynomial is in standard form, we need to ensure that the polynomial is written in descending order of the powers of the variables. Let's compare each option:
1. Polynomial: [tex]\(4xy + 3x^3y^5 - 2x^5y^7 + 4x^7y^9\)[/tex]
- Terms: [tex]\(4xy\)[/tex], [tex]\(3x^3y^5\)[/tex], [tex]\(-2x^5y^7\)[/tex], [tex]\(4x^7y^9\)[/tex]
- Ordering: Descending order based on the total degree of each term.
- Highest combined degree term is [tex]\(4x^7y^9\)[/tex], followed by [tex]\( -2x^5y^7\)[/tex], then [tex]\(3x^3y^5\)[/tex], and finally [tex]\(4xy\)[/tex].
This polynomial is already in standard form.
2. Polynomial: [tex]\(2x^5y^7 + 7y - 8x^2y^5 - 12xy^2\)[/tex]
- Terms: [tex]\(2x^5y^7\)[/tex], [tex]\(7y\)[/tex], [tex]\(-8x^2y^5\)[/tex], [tex]\(-12xy^2\)[/tex]
- Ordering: Currently, there isn’t a clear order from highest to lowest power.
This polynomial is not in standard form.
3. Polynomial: [tex]\(5x^5 - 9x^2y^2 - 3xy^3 + 6y^5\)[/tex]
- Terms: [tex]\(5x^5\)[/tex], [tex]\(-9x^2y^2\)[/tex], [tex]\(-3xy^3\)[/tex], [tex]\(6y^5\)[/tex]
- Ordering: This polynomial also requires reordering.
This polynomial is not in standard form.
4. Polynomial: [tex]\(7x^7y^2 + 5x^{11}y^5 - 3xy^2 + 2\)[/tex]
- Terms: [tex]\(7x^7y^2\)[/tex], [tex]\(5x^{11}y^5\)[/tex], [tex]\(-3xy^2\)[/tex], [tex]\(2\)[/tex]
- Ordering: To standardize, the highest degree term [tex]\(5x^{11}y^5\)[/tex] should come first.
This polynomial is not in standard form.
Based on this evaluation:
The polynomial [tex]\(4xy + 3x^3y^5 - 2x^5y^7 + 4x^7y^9\)[/tex] is the one in standard form.
