To determine which algebraic expression is a polynomial with a degree of 5, let's analyze each of the given polynomial expressions. The degree of a polynomial is the highest sum of the exponents of the variables in any single term.
1. [tex]\( 3x^5 + 8x^4 y^2 - 9x^3 y^3 - 6y^5 \)[/tex]
- [tex]\( 3x^5 \)[/tex]: Degree is [tex]\(5\)[/tex].
- [tex]\( 8x^4 y^2 \)[/tex]: Degree is [tex]\(4 + 2 = 6\)[/tex].
- [tex]\( 9x^3 y^3 \)[/tex]: Degree is [tex]\(3 + 3 = 6\)[/tex].
- [tex]\( 6y^5 \)[/tex]: Degree is [tex]\(5\)[/tex].
The highest degree is [tex]\(6\)[/tex].
2. [tex]\( 2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4 \)[/tex]
- [tex]\( 2x y^4 \)[/tex]: Degree is [tex]\(1 + 4 = 5\)[/tex].
- [tex]\( 4x^2 y^3 \)[/tex]: Degree is [tex]\(2 + 3 = 5\)[/tex].
- [tex]\( 6x^3 y^2 \)[/tex]: Degree is [tex]\(3 + 2 = 5\)[/tex].
- [tex]\( 7x^4 \)[/tex]: Degree is [tex]\(4\)[/tex].
The highest degree is [tex]\(5\)[/tex].
3. [tex]\( 8y^6 + y^5 - 5x y^3 + 7x^2 y^2 - x^3 y - 6x^4 \)[/tex]
- [tex]\( 8y^6 \)[/tex]: Degree is [tex]\(6\)[/tex].
- [tex]\( y^5 \)[/tex]: Degree is [tex]\(5\)[/tex].
- [tex]\( 5x y^3 \)[/tex]: Degree is [tex]\(1 + 3 = 4\)[/tex].
- [tex]\( 7x^2 y^2 \)[/tex]: Degree is [tex]\(2 + 2 = 4\)[/tex].
- [tex]\( x^3 y \)[/tex]: Degree is [tex]\(3 + 1 = 4\)[/tex].
- [tex]\( 6x^4 \)[/tex]: Degree is [tex]\(4\)[/tex].
The highest degree is [tex]\(6\)[/tex].
4. [tex]\( -6x y^5 + 5x^2 y^3 - x^3 y^2 + 2x^2 y^3 - 3x y^5 \)[/tex]
- [tex]\( -6x y^5 \)[/tex]: Degree is [tex]\(1 + 5 = 6\)[/tex].
- [tex]\( 5x^2 y^3 \)[/tex]: Degree is [tex]\(2 + 3 = 5\)[/tex].
- [tex]\( x^3 y^2 \)[/tex]: Degree is [tex]\(3 + 2 = 5\)[/tex].
- [tex]\( 2x^2 y^3 \)[/tex]: Degree is [tex]\(2 + 3 = 5\)[/tex].
- [tex]\( 3x y^5 \)[/tex]: Degree is [tex]\(1 + 5 = 6\)[/tex].
The highest degree is [tex]\(6\)[/tex].
After analyzing all the given expressions, we find that only the second expression has a maximum degree of [tex]\(5\)[/tex]:
[tex]\[ 2x y^4 + 4x^2 y^3 - 6x^3 y^2 - 7x^4 \][/tex]
So, the algebraic expression that is a polynomial with a degree of [tex]\(5\)[/tex] is the second expression.
