To find $P(-1)$ for the polynomial $P(x) = x^3 + 4x^2 + 7$ using the Remainder Theorem, we follow these steps:

The Remainder Theorem states that if a polynomial $P(x)$ is divided by $x - a$, the remainder of this division is $P(a)$. Thus, to find $P(-1)$, we can simply substitute $x = -1$ into the polynomial.

Given: $P(x) = x^3 + 4x^2 + 7$

Substitute $x = -1$: $P(-1) = (-1)^3 + 4(-1)^2 + 7$

Now, calculate each term: $(-1)^3 = -1$ $4(-1)^2 = 4 \cdot 1 = 4$ $7 \text{ remains as it is}$

Combine the results: $P(-1) = -1 + 4 + 7$ $P(-1) = 10$

Therefore, $P(-1) = 10$.

Would you like any more details or have any questions?

Here are some related questions for further exploration:

1. What is the Remainder Theorem?
2. How can you use the Remainder Theorem to determine the roots of a polynomial?
3. What is the value of $P(2)$ for the same polynomial?
4. How would you apply the Factor Theorem to $P(x) = x^3 + 4x^2 + 7$?
5. What are synthetic division and how does it relate to the Remainder Theorem?
6. How do you factorize a cubic polynomial?
7. What are the roots of $P(x) = x^3 + 4x^2 + 7$?
8. How do you graph the polynomial $P(x) = x^3 + 4x^2 + 7$?

Tip: Always check if you can factorize a polynomial to simplify the evaluation process.