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.