Given the function  
P
(
x
)
=
(
x
−
4
)
(
x
+
2
)
(
x
−
6
)
 :

the coordinates of its  
P
 -intercept are 


the coordinates of its  
x
 -intercepts are 

Given the function:

\[
P(x) = (x - 4)(x + 2)(x - 6)
\]

### 1. **P-intercept (y-intercept)**

The **P-intercept** or **y-intercept** is the point where the graph of the function crosses the y-axis. This occurs when \(x = 0\).

To find the P-intercept, substitute \(x = 0\) into the function:

\[
P(0) = (0 - 4)(0 + 2)(0 - 6) = (-4)(2)(-6) = 48
\]

So, the coordinates of the P-intercept are:
\[
(0, 48)
\]

### 2. **x-intercepts**

The **x-intercepts** are the points where the graph crosses the x-axis, which occurs when \(P(x) = 0\).

Set the function equal to 0:
\[
(x - 4)(x + 2)(x - 6) = 0
\]

The solutions occur when any of the factors equals zero:
\[
x - 4 = 0 \quad \Rightarrow \quad x = 4
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]

Thus, the coordinates of the x-intercepts are:
\[
(4, 0), (-2, 0), (6, 0)
\]

### Summary:
- **P-intercept (y-intercept):** \( (0, 48) \)
- **x-intercepts:** \( (4, 0), (-2, 0), (6, 0) \)

Do you want more details or have any questions?  
Here are some related questions:

1. How can we determine the symmetry of this cubic function?
2. What is the behavior of the graph near each x-intercept?
3. How does the sign of the leading coefficient affect the shape of the graph?
4. What are the local maximum and minimum points of this function?
5. Can this function be factored further or simplified?

**Tip:** The degree of a polynomial gives a good indication of how many x-intercepts and turning points the graph might have.

Given the function P(x) = (x − 4)(x + 2)(x − 6), the coordinates of its P-intercept are and the coordinates of its x-intercepts are

Discover how to find the P-intercept and x-intercepts for the polynomial function P(x) = (x − 4)(x + 2)(x − 6). This detailed explanation covers key algebraic concepts like polynomial functions, the zero product property, and intercepts, making it ideal for high school students (Grades 9-12).

Math Problem Statement

Given the function
P ( x )

Solution

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