Divide the given polynomial using the long division method. P(X) =3x²-8x³+9x²-14x+8,D(x)=x-2

To divide the polynomial \( P(x) = -8x^3 + 3x^2 - 14x + 8 \) by \( D(x) = x - 2 \) using the long division method, follow these steps:

1. **Set up the division:**

\[
\begin{array}{r|rrr}
      & -8x^3 & + 3x^2 & - 14x & + 8 \\
x - 2 & 
\end{array}
\]

2. **Divide the first term of the dividend by the first term of the divisor:**

\[
\frac{-8x^3}{x} = -8x^2
\]

Place \(-8x^2\) above the division bar:

\[
\begin{array}{r|rrrr}
     & -8x^2 & + 3x & - 14x & + 8 \\
x - 2 & -8x^2 
\end{array}
\]

3. **Multiply \(-8x^2\) by \( x - 2 \) and subtract:**

\[
(-8x^2)(x - 2) = -8x^3 + 16x^2
\]

\[
\begin{array}{r|rrrr}
     & -8x^2 & + 3x & - 14x & + 8 \\
x - 2 & -8x^2 \\
\hline
     & 0 & -13x^2 & -14x & + 8
\end{array}
\]

4. **Repeat the process with the new polynomial \( -13x^2 - 14x + 8 \):**

\[
\frac{-13x^2}{x} = -13x
\]

Place \(-13x\) above the division bar:

\[
\begin{array}{r|rrrr}
     & -8x^2 & - 13x & - 14x & + 8 \\
x - 2 & -8x^2 & -13x
\end{array}
\]

5. **Multiply \(-13x\) by \( x - 2 \) and subtract:**

\[
(-13x)(x - 2) = -13x^2 + 26x
\]

\[
\begin{array}{r|rrrr}
     & -8x^2 & -13x & - 14x & + 8 \\
x - 2 & -8x^2 & -13x \\
\hline
     & 0 & 0 & -40x & + 8
\end{array}
\]

6. **Repeat the process with the new polynomial \( -40x + 8 \):**

\[
\frac{-40x}{x} = -40
\]

Place \(-40\) above the division bar:

\[
\begin{array}{r|rrrr}
     & -8x^2 & -13x & -40 & + 8 \\
x - 2 & -8x^2 & -13x & -40
\end{array}
\]

7. **Multiply \(-40\) by \( x - 2 \) and subtract:**

\[
(-40)(x - 2) = -40x + 80
\]

\[
\begin{array}{r|rrrr}
     & -8x^2 & -13x & -40 & + 8 \\
x - 2 & -8x^2 & -13x & -40 \\
\hline
     & 0 & 0 & 0 & -72
\end{array}
\]

Therefore, the quotient is \(-8x^2 - 13x - 40\) and the remainder is \( -72 \). 

The result of the division is:

\[
\frac{-8x^3 + 3x^2 - 14x + 8}{x - 2} = -8x^2 - 13x - 40 + \frac{-72}{x - 2}
\]

Would you like any further details or have any questions about this process?

### Related Questions:
1. How do you verify the result of polynomial long division?
2. What are the steps to perform synthetic division for the given polynomial?
3. How can you check if \( x - 2 \) is a factor of \( -8x^3 + 3x^2 - 14x + 8 \)?
4. What is the significance of the remainder in polynomial division?
5. How do you handle polynomial division when the divisor is not linear?
6. Can you explain the relationship between polynomial division and the factor theorem?
7. How do you interpret the quotient and remainder in polynomial division in real-world contexts?
8. What are some common mistakes to avoid in polynomial long division?

### Tip:
Always align terms of the same degree vertically when performing polynomial long division to avoid mistakes.

Learn how to divide the polynomial -8x^3 + 3x^2 - 14x + 8 by x - 2 using the long division method. Step-by-step solution with detailed instructions and example calculation. Suitable for advanced high school students and beyond.

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