Sure! Let's determine the degree of each polynomial step by step:
1. Polynomial: [tex]\( -w^0 \)[/tex]
- Degree:
- The term [tex]\( w^0 \)[/tex] represents a constant, which is not influenced by any variable.
- Therefore, the degree of this term is 0.
- Result: Degree [tex]\( 0 \)[/tex]
2. Polynomial: [tex]\( 36x^8 - 2x^9 + 8x^7 \)[/tex]
- Degree:
- The degrees of the individual terms are [tex]\( 8 \)[/tex] for [tex]\( 36x^8 \)[/tex], [tex]\( 9 \)[/tex] for [tex]\( -2x^9 \)[/tex], and [tex]\( 7 \)[/tex] for [tex]\( 8x^7 \)[/tex].
- The highest exponent among these terms is 9.
- Result: Degree [tex]\( 9 \)[/tex]
3. Polynomial: [tex]\( \frac{rs}{3} + \frac{xy}{7} \)[/tex]
- Degree:
- The degree of a product of variables is the sum of the degrees of the variables in the product.
- For [tex]\( \frac{rs}{3} \)[/tex], the degree is [tex]\( 1 \)[/tex] for [tex]\( r \)[/tex] plus [tex]\( 1 \)[/tex] for [tex]\( s \)[/tex], which gives [tex]\( 1 + 1 = 2 \)[/tex].
- For [tex]\( \frac{xy}{7} \)[/tex], the degree is [tex]\( 1 \)[/tex] for [tex]\( x \)[/tex] plus [tex]\( 1 \)[/tex] for [tex]\( y \)[/tex], which gives [tex]\( 1 + 1 = 2 \)[/tex].
- The highest degree among the terms in this polynomial is [tex]\( 2 \)[/tex].
- Result: Degree [tex]\( 2 \)[/tex]
4. Polynomial: [tex]\( 3x^4 + 9x^3 - 4x^5 + 1 \)[/tex]
- Degree:
- The degrees of the individual terms are [tex]\( 4 \)[/tex] for [tex]\( 3x^4 \)[/tex], [tex]\( 3 \)[/tex] for [tex]\( 9x^3 \)[/tex], [tex]\( 5 \)[/tex] for [tex]\( -4x^5 \)[/tex], and [tex]\( 0 \)[/tex] for the constant term [tex]\( 1 \)[/tex].
- The highest exponent among these terms is 5.
- Result: Degree [tex]\( 5 \)[/tex]
To summarize, the degrees of the given polynomials are:
1. Degree [tex]\( 0 \)[/tex]
2. Degree [tex]\( 9 \)[/tex]
3. Degree [tex]\( 2 \)[/tex]
4. Degree [tex]\( 5 \)[/tex]
