Let's analyze the polynomial [tex]\( g(x) = -\frac{1}{2} x^2 - 5 x + 7 \)[/tex]. Here is a detailed step-by-step solution:
1. Identify the Leading Term:
- The leading term of a polynomial is the term with the highest power of [tex]\(x\)[/tex]. In this polynomial, the highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex], so the leading term is [tex]\( -\frac{1}{2} x^2 \)[/tex].
2. Determine the Leading Coefficient:
- The leading coefficient is the coefficient of the leading term. In this case, the leading term is [tex]\( -\frac{1}{2} x^2 \)[/tex], so the leading coefficient is [tex]\( -\frac{1}{2} \)[/tex].
3. Find the Degree of the Polynomial:
- The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the polynomial. Here, the highest power of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex], which has a degree of 2.
4. Classify the Polynomial:
- A polynomial is classified based on its degree:
- Degree 0: Constant
- Degree 1: Linear
- Degree 2: Quadratic
- Degree 3: Cubic
- Degree 4: Quartic
- Since the degree of this polynomial is 2, it is classified as a quadratic polynomial.
To summarize:
- Leading Term: [tex]\( -\frac{1}{2} x^2 \)[/tex]
- Leading Coefficient: [tex]\( -\frac{1}{2} \)[/tex]
- Degree: 2
- Classification: Quadratic
Thus, the polynomial [tex]\( g(x) = -\frac{1}{2} x^2 - 5 x + 7 \)[/tex] has a leading term of [tex]\( -\frac{1}{2} x^2 \)[/tex], a leading coefficient of [tex]\( -\frac{1}{2} \)[/tex], a degree of 2, and is classified as a quadratic polynomial.
