To find the intercepts of the function [tex]\( f(x) = -3x^3 - 21x^2 - 30x \)[/tex], we will follow these steps:
### Finding the [tex]\(x\)[/tex]-Intercepts
The [tex]\(x\)[/tex]-intercepts are the points where the graph of the function crosses the [tex]\(x\)[/tex]-axis. This happens when [tex]\( f(x) = 0 \)[/tex]. Therefore, we need to solve the equation:
[tex]\[
-3x^3 - 21x^2 - 30x = 0
\][/tex]
#### Step 1: Factoring the equation
First, we can factor out the greatest common factor from the polynomial:
[tex]\[
-3x(x^2 + 7x + 10) = 0
\][/tex]
This gives us two factors to solve:
[tex]\[
-3x = 0 \quad \text{and} \quad x^2 + 7x + 10 = 0
\][/tex]
#### Step 2: Solving the linear factor
The equation [tex]\(-3x = 0\)[/tex] is straightforward to solve:
[tex]\[
x = 0
\][/tex]
#### Step 3: Solving the quadratic factor
Now, we need to solve the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex]. We can factor this quadratic as:
[tex]\[
(x + 2)(x + 5) = 0
\][/tex]
Setting each factor equal to zero gives us:
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
[tex]\[
x + 5 = 0 \quad \Rightarrow \quad x = -5
\][/tex]
So, the [tex]\(x\)[/tex]-intercepts of the function are:
[tex]\[
x = 0, \, x = -2, \, \text{and} \, x = -5
\][/tex]
### Finding the [tex]\(y\)[/tex]-Intercept
The [tex]\(y\)[/tex]-intercept is the point where the graph of the function crosses the [tex]\(y\)[/tex]-axis. This happens when [tex]\( x = 0 \)[/tex]. To find the [tex]\(y\)[/tex]-intercept, we evaluate the function at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = -3(0)^3 - 21(0)^2 - 30(0) = 0
\][/tex]
So, the [tex]\(y\)[/tex]-intercept of the function is:
[tex]\[
f(0) = 0
\][/tex]
### Summary
The intercepts of the function [tex]\( f(x) = -3x^3 - 21x^2 - 30x \)[/tex] are:
- [tex]\(x\)[/tex]-intercepts: [tex]\((0, 0), (-2, 0), (-5, 0)\)[/tex]
- [tex]\(y\)[/tex]-intercept: [tex]\((0, 0)\)[/tex]
These points indicate where the graph crosses the [tex]\(x\)[/tex]-axis and [tex]\(y\)[/tex]-axis.
