Sure! Let's determine the degrees of the given polynomials step-by-step.
### (i) [tex]\( 6x^2y - 2x^2y^3 - xy + 7 \)[/tex]
To find the degree of the polynomial, we need to determine the highest degree of any term in the polynomial. The degree of a term is the sum of the powers of the variables in that term.
1. [tex]\( 6x^2y \)[/tex]:
- Degree: [tex]\( 2 \)[/tex] (from [tex]\( x^2 \)[/tex]) [tex]\( + 1 \)[/tex] (from [tex]\( y \)[/tex]) [tex]\( = 3 \)[/tex]
2. [tex]\( -2x^2y^3 \)[/tex]:
- Degree: [tex]\( 2 \)[/tex] (from [tex]\( x^2 \)[/tex]) [tex]\( + 3 \)[/tex] (from [tex]\( y^3 \)[/tex]) [tex]\( = 5 \)[/tex]
3. [tex]\( -xy \)[/tex]:
- Degree: [tex]\( 1 \)[/tex] (from [tex]\( x \)[/tex]) [tex]\( + 1 \)[/tex] (from [tex]\( y \)[/tex]) [tex]\( = 2 \)[/tex]
4. [tex]\( 7 \)[/tex]:
- Degree: This is a constant term, so it has a degree of [tex]\( 0 \)[/tex]
The highest degree among these terms is [tex]\( 5 \)[/tex]. Therefore, the degree of the polynomial [tex]\( 6x^2 y - 2x^2 y^3 - xy + 7 \)[/tex] is [tex]\( 5 \)[/tex].
### (ii) [tex]\( xy z - x^3 + 6x^2 \)[/tex]
Similarly, we find the degree of each term in the polynomial:
1. [tex]\( xyz \)[/tex]:
- Degree: [tex]\( 1 \)[/tex] (from [tex]\( x \)[/tex]) [tex]\( + 1 \)[/tex] (from [tex]\( y \)[/tex]) [tex]\( + 1 \)[/tex] (from [tex]\( z \)[/tex]) [tex]\( = 3 \)[/tex]
2. [tex]\( -x^3 \)[/tex]:
- Degree: [tex]\( 3 \)[/tex] (from [tex]\( x^3 \)[/tex])
3. [tex]\( 6x^2 \)[/tex]:
- Degree: [tex]\( 2 \)[/tex] (from [tex]\( x^2 \)[/tex])
The highest degree among these terms is [tex]\( 3 \)[/tex]. Therefore, the degree of the polynomial [tex]\( xyz - x^3 + 6x^2 \)[/tex] is [tex]\( 3 \)[/tex].
### Summary
- The degree of the polynomial [tex]\( 6x^2 y - 2x^2 y^3 - xy + 7 \)[/tex] is [tex]\( 5 \)[/tex].
- The degree of the polynomial [tex]\( xyz - x^3 + 6x^2 \)[/tex] is [tex]\( 3 \)[/tex].
