Answer :
To find the leading coefficient of the polynomial function, we need to identify the term with the highest degree and then determine the coefficient of that term.
Let's look at the polynomial function given:
[tex]\[ F(x) = \frac{1}{3} x^3 + 8 x^4 - 5 x - 19 x^2 \][/tex]
1. Identify the degrees of the terms:
- The term [tex]\(\frac{1}{3} x^3\)[/tex] has a degree of 3.
- The term [tex]\(8 x^4\)[/tex] has a degree of 4.
- The term [tex]\(-5 x\)[/tex] has a degree of 1.
- The term [tex]\(-19 x^2\)[/tex] has a degree of 2.
2. Find the term with the highest degree:
- Among the terms [tex]\(x^3\)[/tex], [tex]\(x^4\)[/tex], [tex]\(x\)[/tex], and [tex]\(x^2\)[/tex], the highest degree is 4, which corresponds to the term [tex]\(8 x^4\)[/tex].
3. Determine the leading coefficient:
- The leading coefficient is the coefficient of the term with the highest degree.
- In this case, the coefficient of [tex]\(8 x^4\)[/tex] is 8.
Therefore, the leading coefficient of the polynomial function [tex]\(F(x)\)[/tex] is 8. Hence, the correct choice is:
C. 8
Let's look at the polynomial function given:
[tex]\[ F(x) = \frac{1}{3} x^3 + 8 x^4 - 5 x - 19 x^2 \][/tex]
1. Identify the degrees of the terms:
- The term [tex]\(\frac{1}{3} x^3\)[/tex] has a degree of 3.
- The term [tex]\(8 x^4\)[/tex] has a degree of 4.
- The term [tex]\(-5 x\)[/tex] has a degree of 1.
- The term [tex]\(-19 x^2\)[/tex] has a degree of 2.
2. Find the term with the highest degree:
- Among the terms [tex]\(x^3\)[/tex], [tex]\(x^4\)[/tex], [tex]\(x\)[/tex], and [tex]\(x^2\)[/tex], the highest degree is 4, which corresponds to the term [tex]\(8 x^4\)[/tex].
3. Determine the leading coefficient:
- The leading coefficient is the coefficient of the term with the highest degree.
- In this case, the coefficient of [tex]\(8 x^4\)[/tex] is 8.
Therefore, the leading coefficient of the polynomial function [tex]\(F(x)\)[/tex] is 8. Hence, the correct choice is:
C. 8
