To find the degree and leading coefficient of the polynomial
[tex]\[ 10 - w^6 - 3w - 9w^2, \][/tex]
we need to analyze its terms. Here is a detailed, step-by-step solution:
1. Identify the terms of the polynomial:
[tex]\[ 10, \, -w^6, \, -3w, \, -9w^2 \][/tex]
2. Determine the exponent (or power) of the variable [tex]\( w \)[/tex] in each term:
- [tex]\( 10 \)[/tex]: Since this is a constant term, it has an exponent of 0 (i.e., [tex]\( 10 = 10w^0 \)[/tex]).
- [tex]\( -w^6 \)[/tex]: The exponent here is [tex]\( 6 \)[/tex].
- [tex]\( -3w \)[/tex]: The exponent here is [tex]\( 1 \)[/tex].
- [tex]\( -9w^2 \)[/tex]: The exponent here is [tex]\( 2 \)[/tex].
3. Find the term with the highest exponent:
Among the exponents identified (0, 6, 1, 2), the highest is [tex]\( 6 \)[/tex]. Hence, the term with the highest exponent is [tex]\( -w^6 \)[/tex].
4. Determine the degree of the polynomial:
The degree of a polynomial is the highest exponent of the variable. Therefore, the degree of the polynomial [tex]\( 10 - w^6 - 3w - 9w^2 \)[/tex] is [tex]\( 6 \)[/tex].
5. Find the coefficient of the term with the highest exponent:
The term [tex]\( -w^6 \)[/tex] has a coefficient of [tex]\( -1 \)[/tex]. This is because [tex]\( -w^6 = -1 \cdot w^6 \)[/tex].
So, summarizing the results:
- The degree of the polynomial is [tex]\( 6 \)[/tex].
- The leading coefficient of the polynomial is [tex]\( -1 \)[/tex].
Thus, we can write:
[tex]\[
\text{Degree: } 6
\][/tex]
[tex]\[
\text{Leading coefficient: } -1
\][/tex]
