Find the quotient when (x³-6x²+2x+8) is divided by (x-3)

To find the quotient when \( x^3 - 6x^2 + 2x + 8 \) is divided by \( x - 3 \), we can use polynomial long division or synthetic division. Here, let's use synthetic division as it is quicker for this type of problem.

### Steps for Synthetic Division:

1. **Set up the synthetic division** using the zero of the divisor \( x - 3 \), which is \( 3 \).
2. **Write down the coefficients** of the dividend: \( 1, -6, 2, 8 \).
3. **Perform synthetic division** as follows:

\[
\begin{array}{r|rrrr}
3 & 1 & -6 & 2 & 8 \\
  &   & 3 & -9 & -21 \\
\hline
  & 1 & -3 & -7 & -13 \\
\end{array}
\]

### Explanation of Steps:

- **Step 1**: Bring down the first coefficient \(1\).
- **Step 2**: Multiply \(3\) (from the divisor \(x - 3\)) by the number below the line \(1\) and place it under the next coefficient: \(3 \times 1 = 3\).
- **Step 3**: Add this value to the next coefficient: \(-6 + 3 = -3\).
- **Step 4**: Repeat the multiplication: \(3 \times -3 = -9\).
- **Step 5**: Add to the next coefficient: \(2 + (-9) = -7\).
- **Step 6**: Multiply again: \(3 \times -7 = -21\).
- **Step 7**: Add to the last coefficient: \(8 + (-21) = -13\).

### Result:

- The quotient is \( x^2 - 3x - 7 \).
- The remainder is \(-13\).

### Final Answer:

\[
\frac{x^3 - 6x^2 + 2x + 8}{x - 3} = x^2 - 3x - 7 - \frac{13}{x - 3}
\]

Would you like a detailed explanation of synthetic division or have any other questions?

### Related Questions:

1. How do you perform polynomial long division?
2. What is the Remainder Theorem, and how does it relate to synthetic division?
3. How can we verify the result of synthetic division?
4. What is the relationship between the roots of a polynomial and synthetic division?
5. How do you handle synthetic division when the divisor is not in the form \(x - c\)?

### Tip:
For synthetic division, always ensure the polynomial is written in standard form, including all terms with zero coefficients if needed.

Explore how to find the quotient when dividing the polynomial x³-6x²+2x+8 by x-3 using synthetic division. Learn key algebraic concepts such as polynomial division and the Remainder Theorem, with detailed steps. Suitable for students in Grades 10-12.

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