Answer :
To determine which polynomial is in standard form, let's break down the process and analyze each polynomial step by step. A polynomial in standard form is one where the terms are arranged in descending order of their total degrees.
The total degree of a term is the sum of the exponents of all the variables in that term.
### Analyzing Polynomial 1:
[tex]\[ P_1 = 3xy + 6x^3 y^2 - 4x^4 y^3 + 19x^7 y^4 \][/tex]
- \(3xy\): Total degree = \(1 + 1 = 2\)
- \(6x^3 y^2\): Total degree = \(3 + 2 = 5\)
- \(4x^4 y^3\): Total degree = \(4 + 3 = 7\)
- \(19x^7 y^4\): Total degree = \(7 + 4 = 11\)
Degrees of terms: \([2, 5, 7, 11]\)
### Analyzing Polynomial 2:
[tex]\[ P_2 = 18x^5 - 7x^2 y - 2xy^2 + 17y^4 \][/tex]
- \(18x^5\): Total degree = \(5\)
- \(-7x^2 y\): Total degree = \(2 + 1 = 3\)
- \(-2xy^2\): Total degree = \(1 + 2 = 3\)
- \(17y^4\): Total degree = \(4\)
Degrees of terms: \([5, 3, 3, 4]\)
### Analyzing Polynomial 3:
[tex]\[ P_3 = x^5 y^5 - 3xy - 11x^2 y^2 + 12 \][/tex]
- \(x^5 y^5\): Total degree = \(5 + 5 = 10\)
- \(-3xy\): Total degree = \(1 + 1 = 2\)
- \(-11x^2 y^2\): Total degree = \(2 + 2 = 4\)
- \(12\): Total degree = \(0\)
Degrees of terms: \([10, 2, 4, 0]\)
### Analyzing Polynomial 4:
[tex]\[ P_4 = 15 + 12xy^2 - 11x^9 y^5 + 5x^7 y^2 \][/tex]
- \(15\): Total degree = \(0\)
- \(12xy^2\): Total degree = \(1 + 2 = 3\)
- \(-11x^9 y^5\): Total degree = \(9 + 5 = 14\)
- \(5x^7 y^2\): Total degree = \(7 + 2 = 9\)
Degrees of terms: \([0, 3, 14, 9]\)
### Conclusion:
By checking if the degrees are in descending order for each polynomial:
- For \( P_1 \): The degrees \([2, 5, 7, 11]\) are in ascending order, not descending.
- For \( P_2 \): The degrees \([5, 3, 3, 4]\) are not in descending order.
- For \( P_3 \): The degrees \([10, 2, 4, 0]\) are not in descending order.
- For \( P_4 \): The degrees \([0, 3, 14, 9]\) are not in descending order.
None of the given polynomials are in standard form. Therefore, the correct answer is that no polynomial is in standard form.
Thus, the result is:
[tex]\[ \boxed{\text{None}} \][/tex]
The total degree of a term is the sum of the exponents of all the variables in that term.
### Analyzing Polynomial 1:
[tex]\[ P_1 = 3xy + 6x^3 y^2 - 4x^4 y^3 + 19x^7 y^4 \][/tex]
- \(3xy\): Total degree = \(1 + 1 = 2\)
- \(6x^3 y^2\): Total degree = \(3 + 2 = 5\)
- \(4x^4 y^3\): Total degree = \(4 + 3 = 7\)
- \(19x^7 y^4\): Total degree = \(7 + 4 = 11\)
Degrees of terms: \([2, 5, 7, 11]\)
### Analyzing Polynomial 2:
[tex]\[ P_2 = 18x^5 - 7x^2 y - 2xy^2 + 17y^4 \][/tex]
- \(18x^5\): Total degree = \(5\)
- \(-7x^2 y\): Total degree = \(2 + 1 = 3\)
- \(-2xy^2\): Total degree = \(1 + 2 = 3\)
- \(17y^4\): Total degree = \(4\)
Degrees of terms: \([5, 3, 3, 4]\)
### Analyzing Polynomial 3:
[tex]\[ P_3 = x^5 y^5 - 3xy - 11x^2 y^2 + 12 \][/tex]
- \(x^5 y^5\): Total degree = \(5 + 5 = 10\)
- \(-3xy\): Total degree = \(1 + 1 = 2\)
- \(-11x^2 y^2\): Total degree = \(2 + 2 = 4\)
- \(12\): Total degree = \(0\)
Degrees of terms: \([10, 2, 4, 0]\)
### Analyzing Polynomial 4:
[tex]\[ P_4 = 15 + 12xy^2 - 11x^9 y^5 + 5x^7 y^2 \][/tex]
- \(15\): Total degree = \(0\)
- \(12xy^2\): Total degree = \(1 + 2 = 3\)
- \(-11x^9 y^5\): Total degree = \(9 + 5 = 14\)
- \(5x^7 y^2\): Total degree = \(7 + 2 = 9\)
Degrees of terms: \([0, 3, 14, 9]\)
### Conclusion:
By checking if the degrees are in descending order for each polynomial:
- For \( P_1 \): The degrees \([2, 5, 7, 11]\) are in ascending order, not descending.
- For \( P_2 \): The degrees \([5, 3, 3, 4]\) are not in descending order.
- For \( P_3 \): The degrees \([10, 2, 4, 0]\) are not in descending order.
- For \( P_4 \): The degrees \([0, 3, 14, 9]\) are not in descending order.
None of the given polynomials are in standard form. Therefore, the correct answer is that no polynomial is in standard form.
Thus, the result is:
[tex]\[ \boxed{\text{None}} \][/tex]
