To determine the leading coefficient of the given polynomial function [tex]\( F(x) = \frac{1}{2} x^2 + 8 - 5x^3 - 19x \)[/tex], follow these steps:
1. Identify the term with the highest power of [tex]\( x \)[/tex]:
- In a polynomial, the leading term is the term with the highest exponent of [tex]\( x \)[/tex].
- The given polynomial is [tex]\( F(x) = \frac{1}{2} x^2 + 8 - 5x^3 - 19x \)[/tex].
2. Identify the exponents of [tex]\( x \)[/tex] in each term:
- The exponent of [tex]\( x \)[/tex] in [tex]\(\frac{1}{2} x^2\)[/tex] is 2.
- The exponent of [tex]\( x \)[/tex] in [tex]\( 8 \)[/tex] is 0 (since [tex]\( 8 \)[/tex] can be thought of as [tex]\( 8x^0 \)[/tex]).
- The exponent of [tex]\( x \)[/tex] in [tex]\( -5x^3 \)[/tex] is 3.
- The exponent of [tex]\( x \)[/tex] in [tex]\( -19x \)[/tex] is 1.
3. Determine which exponent is the highest:
- The highest exponent in the polynomial is 3.
4. Find the coefficient of the term with the highest exponent:
- The term with the highest exponent [tex]\( 3 \)[/tex] is [tex]\( -5x^3 \)[/tex].
- The coefficient of [tex]\( -5x^3 \)[/tex] is [tex]\( -5 \)[/tex].
Therefore, the leading coefficient of the polynomial [tex]\( F(x) = \frac{1}{2} x^2 + 8 - 5x^3 - 19x \)[/tex] is [tex]\( -5 \)[/tex].
The correct answer is:
C. -5
