To determine the leading coefficient of the polynomial [tex]\( F(x) = \frac{1}{4} x^5 + 8 x - 5 x^4 - 19 \)[/tex], follow these steps:
1. Identify the term with the highest degree: The degree of a term in a polynomial is the exponent of [tex]\( x \)[/tex]. In the provided polynomial, the terms and their respective degrees are as follows:
- [tex]\(\frac{1}{4} x^5\)[/tex] has a degree of 5.
- [tex]\(8x\)[/tex] has a degree of 1.
- [tex]\(-5 x^4\)[/tex] has a degree of 4.
- [tex]\(-19\)[/tex] is a constant term with a degree of 0 (since [tex]\( x^0 = 1 \)[/tex]).
2. Find the highest degree: From the identified terms, the term [tex]\(\frac{1}{4} x^5\)[/tex] has the highest degree, which is 5.
3. Determine the leading coefficient: The leading coefficient is the coefficient of the term with the highest degree. In this case, the term with the highest degree is [tex]\(\frac{1}{4} x^5\)[/tex], and its coefficient is [tex]\(\frac{1}{4}\)[/tex].
Thus, the leading coefficient of the polynomial [tex]\( F(x) = \frac{1}{4} x^5 + 8 x - 5 x^4 - 19 \)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
Therefore, the correct answer is:
B. [tex]\(\frac{1}{4}\)[/tex].
