Given the function  
P
(
x
)
=
x
3
−
3
x
2
−
4
x
 , find
its  
y
 -intercept is 
  
its  
x
 -intercepts are  
x
1
=
  
   ,  
x
2
=
  
   and  
x
3
=
  
   with  
x
1
<
x
2
<
x
3
 
When  
x
→
∞
 ,  
y
→
  
∞
  (Input + or - for the answer)
When  
x
→
−
∞
 ,  
y
→
  
∞
  (Input + or - for the answer)

We are given the function \( P(x) = x^3 - 3x^2 - 4x \). We will find the \( y \)-intercept, \( x \)-intercepts, and the end behavior as \( x \to \infty \) and \( x \to -\infty \).

### 1. \( y \)-Intercept:
The \( y \)-intercept occurs when \( x = 0 \). Let's substitute \( x = 0 \) into the function to find \( P(0) \):

\[
P(0) = 0^3 - 3(0)^2 - 4(0) = 0
\]

So, the \( y \)-intercept is \( 0 \).

### 2. \( x \)-Intercepts:
The \( x \)-intercepts occur when \( P(x) = 0 \). To solve for \( x \), we solve the equation:

\[
x^3 - 3x^2 - 4x = 0
\]

Factor out \( x \):

\[
x(x^2 - 3x - 4) = 0
\]

Now solve the quadratic equation \( x^2 - 3x - 4 = 0 \). We can factor this as:

\[
x^2 - 3x - 4 = (x - 4)(x + 1) = 0
\]

So, the solutions are:

\[
x = 0, \quad x = 4, \quad x = -1
\]

Therefore, the \( x \)-intercepts are \( x_1 = -1 \), \( x_2 = 0 \), and \( x_3 = 4 \) (ordered such that \( x_1 < x_2 < x_3 \)).

### 3. End Behavior:
For the end behavior, we look at the leading term \( x^3 \), which dominates the behavior of the function for very large and very small \( x \).

- As \( x \to \infty \), the \( x^3 \) term becomes very large and positive, so \( y \to \infty \). Thus, the answer is \( + \).
- As \( x \to -\infty \), the \( x^3 \) term becomes very large and negative, so \( y \to -\infty \). Thus, the answer is \( - \).

### Final Answers:
- \( y \)-intercept: \( 0 \)
- \( x \)-intercepts: \( x_1 = -1 \), \( x_2 = 0 \), \( x_3 = 4 \)
- When \( x \to \infty \), \( y \to + \infty \)
- When \( x \to -\infty \), \( y \to - \infty \)

Do you need any further details or explanations?

### Related Questions:
1. How do you factor higher-degree polynomials in general?
2. Can you explain the concept of end behavior for polynomial functions?
3. What is the difference between the roots and the y-intercept of a function?
4. How can we confirm if a factorization is correct?
5. What is the relationship between the degree of a polynomial and its graph's shape?

**Tip**: When solving for \( x \)-intercepts, always check if factoring is possible before attempting more complex methods like the quadratic formula.

Given the function P(x) = x^3 − 3x^2 − 4x, find its y-intercept, x-intercepts (x1, x2, x3 such that x1 < x2 < x3), and describe the end behavior as x → ∞ and x → −∞.

Learn how to solve a cubic function P(x) = x^3 − 3x^2 − 4x by finding its y-intercept, x-intercepts (x1, x2, x3) and determining its end behavior as x approaches positive and negative infinity. This solution covers key concepts such as polynomial factoring and cubic function end behavior. Suitable for Grades 10-12.

Math Problem Statement

Given the function
P ( x )

its
x -intercepts are
x 1

,
x 2

and
x 3

Solution

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