Let's analyze the given polynomial:
[tex]\[ g(x) = -\frac{1}{2} x^2 - 5 x + 7. \][/tex]
We'll follow a step-by-step approach to identify the leading term, leading coefficient, and the degree of the polynomial.
### Step 1: Identify the leading term
The leading term of a polynomial is the term with the highest power of [tex]\(x\)[/tex]. In the given polynomial, the terms are:
[tex]\[ -\frac{1}{2} x^2, \quad -5x, \quad 7. \][/tex]
Among these terms, the term with the highest power of [tex]\(x\)[/tex] is [tex]\(-\frac{1}{2} x^2\)[/tex]. Therefore, the leading term of the polynomial is:
[tex]\[ -\frac{1}{2} x^2. \][/tex]
### Step 2: Identify the leading coefficient
The leading coefficient is the coefficient of the leading term. In this case, the coefficient of the leading term [tex]\(-\frac{1}{2} x^2\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex].
### Step 3: Identify the degree of the polynomial
The degree of the polynomial is the highest power of [tex]\(x\)[/tex] in the polynomial. In this case, the highest power of [tex]\(x\)[/tex] in the polynomial [tex]\( g(x) = -\frac{1}{2} x^2 - 5 x + 7 \)[/tex] is 2.
Now, we can summarize our findings:
- The leading term of the polynomial is [tex]\( -\frac{1}{2} x^2 \)[/tex].
- The leading coefficient of the polynomial is [tex]\( -\frac{1}{2} \)[/tex].
- The degree of the polynomial is 2.
