To determine which polynomial is in standard form, we need to ensure that the terms in the polynomial are arranged in descending order of their degrees. The degree of each term is the sum of the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Let's analyze each polynomial:
1. [tex]\( 3xy + 6x^3y^2 - 4x^4y^3 + 19x^7y^4 \)[/tex]
- Degree of [tex]\( 3xy \)[/tex] is [tex]\( 1 + 1 = 2 \)[/tex]
- Degree of [tex]\( 6x^3y^2 \)[/tex] is [tex]\( 3 + 2 = 5 \)[/tex]
- Degree of [tex]\( 4x^4y^3 \)[/tex] is [tex]\( 4 + 3 = 7 \)[/tex]
- Degree of [tex]\( 19x^7y^4 \)[/tex] is [tex]\( 7 + 4 = 11 \)[/tex]
- Ordered correctly by degrees: [tex]\( 19x^7y^4 - 4x^4y^3 + 6x^3y^2 + 3xy \)[/tex]
2. [tex]\( 18x^5 - 7x^2y - 2xy^2 + 17y^4 \)[/tex]
- Degree of [tex]\( 18x^5 \)[/tex] is [tex]\( 5 \)[/tex]
- Degree of [tex]\( 7x^2y \)[/tex] is [tex]\( 2 + 1 = 3 \)[/tex]
- Degree of [tex]\( 2xy^2 \)[/tex] is [tex]\( 1 + 2 = 3 \)[/tex]
- Degree of [tex]\( 17y^4 \)[/tex] is [tex]\( 4 \)[/tex]
- Ordered correctly by degrees: [tex]\( 18x^5 + 17y^4 - 7x^2y - 2xy^2 \)[/tex]
3. [tex]\( x^5y^5 - 3xy - 11x^2y^2 + 12 \)[/tex]
- Degree of [tex]\( x^5y^5 \)[/tex] is [tex]\( 5 + 5 = 10 \)[/tex]
- Degree of [tex]\( 3xy \)[/tex] is [tex]\( 1 + 1 = 2 \)[/tex]
- Degree of [tex]\( 11x^2y^2 \)[/tex] is [tex]\( 2 + 2 = 4 \)[/tex]
- Degree of [tex]\( 12 \)[/tex] is [tex]\( 0 \)[/tex]
- Ordered correctly by degrees: [tex]\( x^5y^5 - 11x^2y^2 - 3xy + 12 \)[/tex]
4. [tex]\( 15 + 12xy^2 - 11x^9y^5 + 5x^7y^2 \)[/tex]
- Degree of [tex]\( 15 \)[/tex] is [tex]\( 0 \)[/tex]
- Degree of [tex]\( 12xy^2 \)[/tex] is [tex]\( 1 + 2 = 3 \)[/tex]
- Degree of [tex]\( 11x^9y^5 \)[/tex] is [tex]\( 9 + 5 = 14 \)[/tex]
- Degree of [tex]\( 5x^7y^2 \)[/tex] is [tex]\( 7 + 2 = 9 \)[/tex]
- Ordered correctly by degrees: [tex]\( -11x^9y^5 + 5x^7y^2 + 12xy^2 + 15 \)[/tex]
After comparing the polynomials, we observe that:
- The polynomial [tex]\( 18x^5 - 7x^2y - 2xy^2 + 17y^4 \)[/tex] is already in standard form because its terms are arranged in descending order of their degrees.
Thus, the polynomial in standard form is:
[tex]\[ 18x^5 - 7x^2y - 2xy^2 + 17y^4 \][/tex]
