To find $P(-2)$ for the polynomial $P(x) = 3x^3 - 6x^2 + 2x - 12$, using synthetic division and the Remainder Theorem, follow these steps:

Step 1: Set up synthetic division

1. Write down the coefficients of the polynomial: $3, -6, 2, -12$.
2. Since we're finding $P(-2)$, use $-2$ as the divisor.

Step 2: Perform synthetic division

• Bring down the first coefficient (3) as is.
• Multiply it by $-2$ and add to the next coefficient:
  • $3 \times (-2) = -6$
  • $-6 + (-6) = -12$
• Continue this process:
  • $-12 \times (-2) = 24$
  • $24 + 2 = 26$
  • $26 \times (-2) = -52$
  • $-52 + (-12) = -64$

Step 3: Interpret the result

• The remainder is the value of $P(-2)$, which is $-64$.

So, $P(-2) = -64$.

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Do you want further details or have any questions?

Here are 5 related questions you might find useful:

1. What is the general process of synthetic division?
2. How does the Remainder Theorem help in finding the value of a polynomial at a specific point?
3. How would you find the roots of the polynomial $3x^3 - 6x^2 + 2x - 12$?
4. What are the benefits of using synthetic division over long division for polynomials?
5. How can you verify the result obtained from synthetic division?

Tip: When using synthetic division, always double-check the sign of the divisor, especially if you are using the Remainder Theorem.