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Sagot :
To determine which polynomial is in standard form, we must inspect each polynomial and ensure that the terms are arranged in descending order of their degrees.
### Polynomial 1:
[tex]\[ 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \][/tex]
Let’s calculate the degrees of each term:
- \( 3xy \) has a degree of \( 1 + 1 = 2 \)
- \( 6x^3y^2 \) has a degree of \( 3 + 2 = 5 \)
- \( -4x^4y^3 \) has a degree of \( 4 + 3 = 7 \)
- \( 19x^7y^4 \) has a degree of \( 7 + 4 = 11 \)
When arranged in descending order of their degrees:
[tex]\[ 19x^7y^4 \rightarrow 11 \][/tex]
[tex]\[ -4x^4y^3 \rightarrow 7 \][/tex]
[tex]\[ 6x^3y^2 \rightarrow 5 \][/tex]
[tex]\[ 3xy \rightarrow 2 \][/tex]
Thus, polynomial 1 is in standard form.
### Polynomial 2:
[tex]\[ 18x^5 - 7x^2y - 2xy^2 + 17y^4 \][/tex]
Let’s calculate the degrees of each term:
- \( 18x^5 \) has a degree of \( 5 \)
- \( -7x^2y \) has a degree of \( 2 + 1 = 3 \)
- \( -2xy^2 \) has a degree of \( 1 + 2 = 3 \)
- \( 17y^4 \) has a degree of \( 4 \)
When arranged in descending order of their degrees:
[tex]\[ 18x^5 \rightarrow 5 \][/tex]
[tex]\[ 17y^4 \rightarrow 4 \][/tex]
[tex]\[ -7x^2y \rightarrow 3 \][/tex]
[tex]\[ -2xy^2 \rightarrow 3 \][/tex]
This polynomial is not in perfect descending order because terms with equal degrees should be grouped together; however, it is considered in descending order by their degrees.
### Polynomial 3:
[tex]\[ x^5y^5 - 3xy - 11x^2y^2 + 12 \][/tex]
Let’s calculate the degrees of each term:
- \( x^5y^5 \) has a degree of \( 5 + 5 = 10 \)
- \( -3xy \) has a degree of \( 1 + 1 = 2 \)
- \( -11x^2y^2 \) has a degree of \( 2 + 2 = 4 \)
- \( 12 \) has a degree of \( 0 \) (Since it is a constant term)
When arranged in descending order of their degrees:
[tex]\[ x^5y^5 \rightarrow 10 \][/tex]
[tex]\[ -11x^2y^2 \rightarrow 4 \][/tex]
[tex]\[ -3xy \rightarrow 2 \][/tex]
[tex]\[ 12 \rightarrow 0 \][/tex]
This polynomial is in standard form.
### Polynomial 4:
[tex]\[ 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \][/tex]
Let’s calculate the degrees of each term:
- \( 15 \) has a degree of \( 0 \) (constant term)
- \( 12xy^2 \) has a degree of \( 1 + 2 = 3 \)
- \( -11x^9y^5 \) has a degree of \( 9 + 5 = 14 \)
- \( 5x^7y^2 \) has a degree of \( 7 + 2 = 9 \)
When arranged in descending order of their degrees:
[tex]\[ -11x^9y^5 \rightarrow 14 \][/tex]
[tex]\[ 5x^7y^2 \rightarrow 9 \][/tex]
[tex]\[ 12xy^2 \rightarrow 3 \][/tex]
[tex]\[ 15 \rightarrow 0 \][/tex]
Thus, polynomial 4 is in standard form.
### Conclusion:
By examining each polynomial, we can confidently say:
- Polynomial 1 and 3 are in standard form.
- Polynomial 2 is in descending degree, but the grouping is not perfect.
- Polynomial 4 is also in standard form.
However, as we stated initially, the solution derived points out polynomial 1 as having the correct standard order with descending degrees.
Therefore, the polynomial in standard form is:
[tex]\[ 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \][/tex]
So, the polynomial in standard form is the first one.
### Polynomial 1:
[tex]\[ 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \][/tex]
Let’s calculate the degrees of each term:
- \( 3xy \) has a degree of \( 1 + 1 = 2 \)
- \( 6x^3y^2 \) has a degree of \( 3 + 2 = 5 \)
- \( -4x^4y^3 \) has a degree of \( 4 + 3 = 7 \)
- \( 19x^7y^4 \) has a degree of \( 7 + 4 = 11 \)
When arranged in descending order of their degrees:
[tex]\[ 19x^7y^4 \rightarrow 11 \][/tex]
[tex]\[ -4x^4y^3 \rightarrow 7 \][/tex]
[tex]\[ 6x^3y^2 \rightarrow 5 \][/tex]
[tex]\[ 3xy \rightarrow 2 \][/tex]
Thus, polynomial 1 is in standard form.
### Polynomial 2:
[tex]\[ 18x^5 - 7x^2y - 2xy^2 + 17y^4 \][/tex]
Let’s calculate the degrees of each term:
- \( 18x^5 \) has a degree of \( 5 \)
- \( -7x^2y \) has a degree of \( 2 + 1 = 3 \)
- \( -2xy^2 \) has a degree of \( 1 + 2 = 3 \)
- \( 17y^4 \) has a degree of \( 4 \)
When arranged in descending order of their degrees:
[tex]\[ 18x^5 \rightarrow 5 \][/tex]
[tex]\[ 17y^4 \rightarrow 4 \][/tex]
[tex]\[ -7x^2y \rightarrow 3 \][/tex]
[tex]\[ -2xy^2 \rightarrow 3 \][/tex]
This polynomial is not in perfect descending order because terms with equal degrees should be grouped together; however, it is considered in descending order by their degrees.
### Polynomial 3:
[tex]\[ x^5y^5 - 3xy - 11x^2y^2 + 12 \][/tex]
Let’s calculate the degrees of each term:
- \( x^5y^5 \) has a degree of \( 5 + 5 = 10 \)
- \( -3xy \) has a degree of \( 1 + 1 = 2 \)
- \( -11x^2y^2 \) has a degree of \( 2 + 2 = 4 \)
- \( 12 \) has a degree of \( 0 \) (Since it is a constant term)
When arranged in descending order of their degrees:
[tex]\[ x^5y^5 \rightarrow 10 \][/tex]
[tex]\[ -11x^2y^2 \rightarrow 4 \][/tex]
[tex]\[ -3xy \rightarrow 2 \][/tex]
[tex]\[ 12 \rightarrow 0 \][/tex]
This polynomial is in standard form.
### Polynomial 4:
[tex]\[ 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \][/tex]
Let’s calculate the degrees of each term:
- \( 15 \) has a degree of \( 0 \) (constant term)
- \( 12xy^2 \) has a degree of \( 1 + 2 = 3 \)
- \( -11x^9y^5 \) has a degree of \( 9 + 5 = 14 \)
- \( 5x^7y^2 \) has a degree of \( 7 + 2 = 9 \)
When arranged in descending order of their degrees:
[tex]\[ -11x^9y^5 \rightarrow 14 \][/tex]
[tex]\[ 5x^7y^2 \rightarrow 9 \][/tex]
[tex]\[ 12xy^2 \rightarrow 3 \][/tex]
[tex]\[ 15 \rightarrow 0 \][/tex]
Thus, polynomial 4 is in standard form.
### Conclusion:
By examining each polynomial, we can confidently say:
- Polynomial 1 and 3 are in standard form.
- Polynomial 2 is in descending degree, but the grouping is not perfect.
- Polynomial 4 is also in standard form.
However, as we stated initially, the solution derived points out polynomial 1 as having the correct standard order with descending degrees.
Therefore, the polynomial in standard form is:
[tex]\[ 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \][/tex]
So, the polynomial in standard form is the first one.
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