To determine the leading coefficient of the polynomial [tex]\( F(x)=\frac{1}{3} x^3+8 x^4-5 x-19 x^2 \)[/tex], follow these steps:
1. Identify the terms and their degrees: The polynomial [tex]\(F(x)\)[/tex] can be broken down into its individual terms:
- [tex]\(\frac{1}{3} x^3\)[/tex] which has a degree of 3,
- [tex]\(8 x^4\)[/tex] which has a degree of 4,
- [tex]\(-5 x\)[/tex] which has a degree of 1,
- [tex]\(-19 x^2\)[/tex] which has a degree of 2.
2. Determine the highest degree term: Among these terms, the term with the highest degree is [tex]\(8 x^4\)[/tex], since 4 is the highest exponent (degree) in the polynomial.
3. Find the leading coefficient: The leading coefficient is the coefficient of the term with the highest degree. For [tex]\(8 x^4\)[/tex], the coefficient is 8.
So, the leading coefficient of the polynomial [tex]\( F(x)=\frac{1}{3} x^3+8 x^4-5 x-19 x^2 \)[/tex] is [tex]\( \boxed{8} \)[/tex].
