Answer :
To determine the leading coefficient of the polynomial function [tex]\( F(x)=\frac{1}{3} x^3+8 x^4-5 x-19 x^2 \)[/tex], follow these steps:
1. Identify and Rewrite the Polynomial Terms:
Rewrite the polynomial so that it is in standard form, with terms ordered by the descending powers of [tex]\( x \)[/tex]:
[tex]\[
F(x) = 8x^4 + \frac{1}{3}x^3 - 19x^2 - 5x
\][/tex]
2. Locate the Term with the Highest Power:
In the polynomial function, identify the term with the highest exponent (largest power of [tex]\( x \)[/tex]). Here, the terms and their exponents are:
- [tex]\( 8x^4 \)[/tex] (highest exponent: 4)
- [tex]\( \frac{1}{3}x^3 \)[/tex]
- [tex]\( -19x^2 \)[/tex]
- [tex]\( -5x \)[/tex]
3. Determine the Leading Coefficient:
The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. Here, the term [tex]\( 8x^4 \)[/tex] has the highest power (4), and its coefficient is [tex]\( 8 \)[/tex].
Thus, the leading coefficient of the polynomial [tex]\( F(x) \)[/tex] is:
Answer: 8 (Option B)
1. Identify and Rewrite the Polynomial Terms:
Rewrite the polynomial so that it is in standard form, with terms ordered by the descending powers of [tex]\( x \)[/tex]:
[tex]\[
F(x) = 8x^4 + \frac{1}{3}x^3 - 19x^2 - 5x
\][/tex]
2. Locate the Term with the Highest Power:
In the polynomial function, identify the term with the highest exponent (largest power of [tex]\( x \)[/tex]). Here, the terms and their exponents are:
- [tex]\( 8x^4 \)[/tex] (highest exponent: 4)
- [tex]\( \frac{1}{3}x^3 \)[/tex]
- [tex]\( -19x^2 \)[/tex]
- [tex]\( -5x \)[/tex]
3. Determine the Leading Coefficient:
The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. Here, the term [tex]\( 8x^4 \)[/tex] has the highest power (4), and its coefficient is [tex]\( 8 \)[/tex].
Thus, the leading coefficient of the polynomial [tex]\( F(x) \)[/tex] is:
Answer: 8 (Option B)
