To find the [tex]\(x\)[/tex]-intercepts and the [tex]\(y\)[/tex]-intercept of the polynomial function [tex]\(g(x) = (x-5)(x+1)(4-x)\)[/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 intersects the [tex]\(x\)[/tex]-axis. These occur when [tex]\(g(x) = 0\)[/tex].
1. Set the function equal to zero:
[tex]\( (x-5)(x+1)(4-x) = 0 \)[/tex]
2. Solve for [tex]\(x\)[/tex] by setting each factor equal to zero:
[tex]\[
x - 5 = 0 \implies x = 5 \\
x + 1 = 0 \implies x = -1 \\
4 - x = 0 \implies x = 4
\][/tex]
3. Therefore, the [tex]\(x\)[/tex]-intercepts are:
[tex]\((5, 0)\)[/tex], [tex]\((-1, 0)\)[/tex], and [tex]\((4, 0)\)[/tex].
### Finding the [tex]\(y\)[/tex]-intercept:
The [tex]\(y\)[/tex]-intercept is the point where the graph of the function intersects the [tex]\(y\)[/tex]-axis. This occurs when [tex]\(x = 0\)[/tex].
1. Substitute [tex]\(x = 0\)[/tex] into the function:
[tex]\[
g(0) = (0 - 5)(0 + 1)(4 - 0)
\][/tex]
2. Simplify the expression:
[tex]\[
g(0) = (-5)(1)(4) = -20
\][/tex]
3. Therefore, the [tex]\(y\)[/tex]-intercept is:
[tex]\((0, -20)\)[/tex]
### Summary of Intercepts:
- [tex]\(x\)[/tex]-intercepts: [tex]\((5, 0)\)[/tex], [tex]\((-1, 0)\)[/tex], [tex]\((4, 0)\)[/tex]
- [tex]\(y\)[/tex]-intercept: [tex]\((0, -20)\)[/tex]
So, the [tex]\(x\)[/tex]-intercepts are [tex]\((5, 0)\)[/tex], [tex]\((-1, 0)\)[/tex], and [tex]\((4, 0)\)[/tex], and the [tex]\(y\)[/tex]-intercept is [tex]\((0, -20)\)[/tex].
