Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial as constant, linear, quadratic, or quartic.

Given:
[tex]\[ g(x) = -\frac{1}{2} x^2 - 5x + 7 \][/tex]

1. The leading term of the polynomial is [tex]\(-\frac{1}{2} x^2\)[/tex].
(Use integers or fractions for any numbers in the expression.)

2. The leading coefficient of the polynomial is [tex]\(-\frac{1}{2}\)[/tex].
(Type an integer or a fraction.)

3. The degree of the polynomial is 2.

4. The polynomial is [tex]\(\boxed{\text{quadratic}}\)[/tex].



Answer :

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.