To find the quotient and remainder when dividing \( P(x) = -x^3 - 3x^2 - x + 6 \) by \( x + 3 \), we can use polynomial long division or synthetic division. Here, we'll use synthetic division for efficiency.

### Synthetic Division

We want to divide \( P(x) \) by \( x + 3 \), so we use \( -3 \) (the zero of \( x + 3 \)) in synthetic division.

1. **Set up the synthetic division:**

   Write down the coefficients of \( P(x) \):
   \[
   -1, -3, -1, 6
   \]

   We place \(-3\) to the left and the coefficients in a row:

   \[
   \begin{array}{r|rrrr}
   -3 & -1 & -3 & -1 & 6 \\
     &    &    &    &   \\
   \end{array}
   \]

2. **Perform the synthetic division steps:**

   - Bring down the first coefficient:
     \[
     \begin{array}{r|rrrr}
     -3 & -1 & -3 & -1 & 6 \\
        &    &    &    &   \\
        & -1 &    &    &   \\
     \end{array}
     \]

   - Multiply \(-3\) by \(-1\) and add to the next coefficient:
     \[
     \begin{array}{r|rrrr}
     -3 & -1 & -3 & -1 & 6 \\
        &    &  3 &    &   \\
        & -1 &  0 &    &   \\
     \end{array}
     \]

   - Multiply \(-3\) by \(0\) and add to the next coefficient:
     \[
     \begin{array}{r|rrrr}
     -3 & -1 & -3 & -1 & 6 \\
        &    &  3 &  0 &   \\
        & -1 &  0 & -1 &   \\
     \end{array}
     \]

   - Multiply \(-3\) by \(-1\) and add to the next coefficient:
     \[
     \begin{array}{r|rrrr}
     -3 & -1 & -3 & -1 & 6 \\
        &    &  3 &  0 &  3 \\
        & -1 &  0 & -1 & 9 \\
     \end{array}
     \]

The final row gives the coefficients of the quotient and the remainder. The quotient is \( -x^2 + 0x - 1 \) and the remainder is \( 9 \).

So,
\[
P(x) = (x + 3)(-x^2 - 1) + 9
\]

### Summary

- **Quotient**: \( -x^2 - 1 \)
- **Remainder**: \( 9 \)

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you perform polynomial long division?
2. Can synthetic division be used for any divisor?
3. What are the steps to verify the quotient and remainder?
4. How is synthetic division different from polynomial long division?
5. How can you use synthetic division to find roots of a polynomial?
6. What is the significance of the remainder in polynomial division?
7. Can you use the Remainder Theorem to check your synthetic division result?
8. How does synthetic division help in graphing polynomials?

**Tip:** Practice synthetic division with various polynomials to become proficient and to understand its efficiency compared to polynomial long division.

Discover how to find the quotient and remainder of polynomial division using synthetic division with a step-by-step guide. This problem involves polynomial long division techniques and applies the Remainder Theorem. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation