The task here involves using the remainder theorem to find the value of $P(3)$ for the polynomial $P(x) = -x^3 + 4x^2 + 8$, as well as the quotient and the remainder from dividing the polynomial by $x - 3$.

Steps to solve the problem:

1. Apply the remainder theorem:
   The remainder theorem tells us that the remainder of the division of a polynomial $P(x)$ by $x - a$ is $P(a)$. Here, we are dividing by $x - 3$, so the remainder will be $P(3)$.

2. Evaluate $P(3)$: $P(3) = -(3)^3 + 4(3)^2 + 8$ Simplifying step by step: $P(3) = -(27) + 4(9) + 8$ $P(3) = -27 + 36 + 8 = 17$

   So, $P(3) = 17$. This value is also the remainder.

3. Find the quotient:
   To find the quotient, we need to perform polynomial division of $P(x) = -x^3 + 4x^2 + 8$ by $x - 3$.

   Using synthetic division:

   • Write the coefficients of $P(x)$: [-1, 4, 0, 8].
   • Use $x - 3$, so the root is 3.

   Perform synthetic division:

   3 | -1   4   0   8
       |      -3   3   9
       ----------------
        -1   1   3   17
   

   The quotient is $-x^2 + x + 3$, and the remainder is 17 (which matches $P(3)$).

Final Answer:

• Quotient: $-x^2 + x + 3$
• Remainder: 17
• $P(3)$ = 17

Would you like more details on synthetic division or have any other questions?

Here are 5 related questions for further practice:

1. How do you perform polynomial division using the long division method?
2. What does the remainder theorem state and how is it useful?
3. Can the quotient be factored further in certain division problems?
4. How do synthetic division and long division compare in efficiency?
5. How would the process change if you divided by a different $x - a$?

Tip: Always verify your quotient by multiplying it back with the divisor and adding the remainder to check if it matches the original polynomial.