Answer :
Sure! Let's go through the solution for the given polynomial function [tex]\( f(x) = 18x^5 + 39x^4 - 100x^3 - 385x^2 - 408x - 144 \)[/tex].
### 1. Degree of [tex]\( f \)[/tex]
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the expression. Here, the term with the highest power is [tex]\( 18x^5 \)[/tex]. Therefore, the degree of [tex]\( f \)[/tex] is 5.
### 2. Leading Coefficient of [tex]\( f \)[/tex]
The leading coefficient is the coefficient of the term with the highest power. In this polynomial, the leading term is [tex]\( 18x^5 \)[/tex] and the leading coefficient is 18.
### 3. End Behavior
The end behavior of a polynomial function is determined by the sign of the leading coefficient and the degree of the polynomial.
- Right Hand End Behavior:
As [tex]\( x \rightarrow +\infty \)[/tex], since the degree is odd and the leading coefficient is positive ([tex]\(+18\)[/tex]), [tex]\( f(x) \rightarrow +\infty \)[/tex].
- Left Hand End Behavior:
As [tex]\( x \rightarrow -\infty \)[/tex], with an odd degree and a positive leading coefficient, [tex]\( f(x) \rightarrow -\infty \)[/tex].
### 4. Zeros of [tex]\( f \)[/tex]
The zeros (roots or solutions) of the polynomial are the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex]. For this polynomial, the zeros are:
- [tex]\( x = -\frac{3}{2} \)[/tex]
- [tex]\( x = -\frac{4}{3} \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 3 \)[/tex]
### 5. Factored Form
The polynomial can be factored as a product of linear factors:
[tex]\[ f(x) = (x - 3)(x + 1)(2x + 3)(3x + 4)^2 \][/tex]
### 6. [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercepts are where the graph of the polynomial crosses the x-axis. These are the zeros of [tex]\( f(x) \)[/tex], which are:
- [tex]\( x = -\frac{3}{2} \)[/tex]
- [tex]\( x = -\frac{4}{3} \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 3 \)[/tex]
### 7. [tex]\( f(x) \)[/tex]-Intercept
The [tex]\( f(x) \)[/tex]-intercept (y-intercept) is the value of [tex]\( f \)[/tex] when [tex]\( x = 0 \)[/tex]. This is the constant term in the polynomial:
- The [tex]\( f(x) \)[/tex]-intercept is [tex]\( -144 \)[/tex].
These details help you understand the behavior and form of the polynomial function graphically and algebraically. Let me know if you have any further questions!
### 1. Degree of [tex]\( f \)[/tex]
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the expression. Here, the term with the highest power is [tex]\( 18x^5 \)[/tex]. Therefore, the degree of [tex]\( f \)[/tex] is 5.
### 2. Leading Coefficient of [tex]\( f \)[/tex]
The leading coefficient is the coefficient of the term with the highest power. In this polynomial, the leading term is [tex]\( 18x^5 \)[/tex] and the leading coefficient is 18.
### 3. End Behavior
The end behavior of a polynomial function is determined by the sign of the leading coefficient and the degree of the polynomial.
- Right Hand End Behavior:
As [tex]\( x \rightarrow +\infty \)[/tex], since the degree is odd and the leading coefficient is positive ([tex]\(+18\)[/tex]), [tex]\( f(x) \rightarrow +\infty \)[/tex].
- Left Hand End Behavior:
As [tex]\( x \rightarrow -\infty \)[/tex], with an odd degree and a positive leading coefficient, [tex]\( f(x) \rightarrow -\infty \)[/tex].
### 4. Zeros of [tex]\( f \)[/tex]
The zeros (roots or solutions) of the polynomial are the values of [tex]\( x \)[/tex] that make [tex]\( f(x) = 0 \)[/tex]. For this polynomial, the zeros are:
- [tex]\( x = -\frac{3}{2} \)[/tex]
- [tex]\( x = -\frac{4}{3} \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 3 \)[/tex]
### 5. Factored Form
The polynomial can be factored as a product of linear factors:
[tex]\[ f(x) = (x - 3)(x + 1)(2x + 3)(3x + 4)^2 \][/tex]
### 6. [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercepts are where the graph of the polynomial crosses the x-axis. These are the zeros of [tex]\( f(x) \)[/tex], which are:
- [tex]\( x = -\frac{3}{2} \)[/tex]
- [tex]\( x = -\frac{4}{3} \)[/tex]
- [tex]\( x = -1 \)[/tex]
- [tex]\( x = 3 \)[/tex]
### 7. [tex]\( f(x) \)[/tex]-Intercept
The [tex]\( f(x) \)[/tex]-intercept (y-intercept) is the value of [tex]\( f \)[/tex] when [tex]\( x = 0 \)[/tex]. This is the constant term in the polynomial:
- The [tex]\( f(x) \)[/tex]-intercept is [tex]\( -144 \)[/tex].
These details help you understand the behavior and form of the polynomial function graphically and algebraically. Let me know if you have any further questions!
