To determine the leading coefficient of a polynomial, we need to identify the term with the highest power of [tex]\(x\)[/tex]. The coefficient of this term is the leading coefficient.
We start by examining the given polynomial:
[tex]\[ F(x) = \frac{1}{2} x^2 + 8 - 5 x^3 - 19 x \][/tex]
1. Identify the degrees of each term in the polynomial:
- The term [tex]\(\frac{1}{2} x^2\)[/tex] has a degree of 2.
- The constant term 8 has a degree of 0 (since it does not depend on [tex]\(x\)[/tex]).
- The term [tex]\(-5 x^3\)[/tex] has a degree of 3.
- The term [tex]\(-19 x\)[/tex] has a degree of 1.
2. Compare the degrees of all the terms:
- The highest degree present in the polynomial is 3, associated with the term [tex]\(-5 x^3\)[/tex].
3. The leading term of the polynomial is the term with the highest degree, which in this case is [tex]\(-5 x^3\)[/tex].
4. The coefficient of this leading term is [tex]\(-5\)[/tex].
Therefore, the leading coefficient of the polynomial [tex]\( F(x) = \frac{1}{2} x^2 + 8 - 5 x^3 - 19 x \)[/tex] is
[tex]\[ \boxed{-5} \][/tex]
