Find the [tex]$x$[/tex]- and [tex]$y$[/tex]-intercepts of the following polynomial function. Express all points as ordered pairs.

[tex]\[ g(x) = (x - 5)(x + 1)(4 - x) \][/tex]

[tex]$x$[/tex]-intercepts: [tex]$\square$[/tex]

[tex]$y$[/tex]-intercept: [tex]$\square$[/tex]



Answer :

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].