https://file.aiquickdraw.com/mathbot/file/1728856151916bd149w2j.jpg

The problem involves evaluating \( P(-1) \) for the polynomial \( P(x) = 3x^2 + 12x + 3 \) using synthetic division and the Remainder Theorem.

### Steps:

1. **Set up synthetic division** with the divisor \( c = -1 \) and the coefficients of the polynomial:
   - The coefficients of \( P(x) = 3x^2 + 12x + 3 \) are \( [3, 12, 3] \).

2. **Synthetic Division Process**:
   - Write down the coefficients: \( 3, 12, 3 \).
   - Perform the synthetic division using \( -1 \) as the divisor.

   \[
   \begin{array}{r|rrr}
   -1 & 3 & 12 & 3 \\
      &   & -3 & -9 \\
   \hline
      & 3 & 9  & -6 \\
   \end{array}
   \]

3. **Interpret the remainder**:
   - The last number, \( -6 \), is the remainder.
   - According to the Remainder Theorem, \( P(-1) \) is equal to the remainder.

Thus, \( P(-1) = -6 \).

Would you like more details on the steps involved or have any questions?

Here are 5 related questions for further practice:
1. What is the Remainder Theorem, and how is it used in polynomial division?
2. How would synthetic division change if \( c = 2 \) instead of \( -1 \)?
3. Can you apply synthetic division to a polynomial of degree higher than 2?
4. What happens if the remainder in synthetic division is 0?
5. How do you verify the result of synthetic division using direct substitution?

**Tip:** Synthetic division is a shortcut for dividing polynomials when the divisor is in the form \( x - c \).

Use synthetic division and the Remainder Theorem to evaluate P(-1) for the polynomial P(x) = 3x^2 + 12x + 3.

This solution explains how to use synthetic division and the Remainder Theorem to evaluate P(-1) for the polynomial P(x) = 3x^2 + 12x + 3. Learn step-by-step how to find the remainder and understand how the theorem applies to polynomials. Perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation