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using synthetic division and the reminder theron for find P(a)

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.

Learn how to find the value of a polynomial P(x) = 3x^3 - 6x^2 + 2x - 12 at a specific point using synthetic division and the Remainder Theorem. Follow a detailed step-by-step solution to compute P(-2) efficiently. Suitable for students studying polynomials and algebra in Grades 10-12.

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