divide x^3+3x^2-2x+5 by x+1 using long divsiion

To divide \( x^3 + 3x^2 - 2x + 5 \) by \( x + 1 \) using long division, follow these steps:

### Step-by-Step Solution

1. **Set up the division:**
   \[
   \begin{array}{r|rrrr}
       x+1 & x^3 & +3x^2 & -2x & +5 \\
       \hline
         & x^2 & 2x & -4 & \\
   \end{array}
   \]

2. **Divide the first term of the dividend \( x^3 \) by the first term of the divisor \( x \):**
   \[
   \frac{x^3}{x} = x^2
   \]
   Write \( x^2 \) above the line.

3. **Multiply \( x^2 \) by the entire divisor \( x+1 \):**
   \[
   x^2 \cdot (x + 1) = x^3 + x^2
   \]

4. **Subtract the result from the original polynomial:**
   \[
   (x^3 + 3x^2 - 2x + 5) - (x^3 + x^2) = 2x^2 - 2x + 5
   \]
   Update the polynomial:
   \[
   \begin{array}{r|rrrr}
       x+1 & x^3 & +3x^2 & -2x & +5 \\
       \hline
         & x^2 & 2x & -4 & \\
         & -(x^3+x^2) & & & \\
       \hline
         & & 2x^2 & -2x & +5 \\
   \end{array}
   \]

5. **Divide the new leading term \( 2x^2 \) by the first term of the divisor \( x \):**
   \[
   \frac{2x^2}{x} = 2x
   \]
   Write \( 2x \) above the line.

6. **Multiply \( 2x \) by the entire divisor \( x+1 \):**
   \[
   2x \cdot (x + 1) = 2x^2 + 2x
   \]

7. **Subtract the result from the current polynomial:**
   \[
   (2x^2 - 2x + 5) - (2x^2 + 2x) = -4x + 5
   \]
   Update the polynomial:
   \[
   \begin{array}{r|rrrr}
       x+1 & x^3 & +3x^2 & -2x & +5 \\
       \hline
         & x^2 & +2x & -4 & \\
         & -(x^3+x^2) & & & \\
       \hline
         & & 2x^2 & -2x & +5 \\
         & -(2x^2 + 2x) & & & \\
       \hline
         & & & -4x & +5 \\
   \end{array}
   \]

8. **Divide the new leading term \( -4x \) by the first term of the divisor \( x \):**
   \[
   \frac{-4x}{x} = -4
   \]
   Write \( -4 \) above the line.

9. **Multiply \( -4 \) by the entire divisor \( x+1 \):**
   \[
   -4 \cdot (x + 1) = -4x - 4
   \]

10. **Subtract the result from the current polynomial:**
    \[
    (-4x + 5) - (-4x - 4) = 9
    \]
    Update the polynomial:
    \[
    \begin{array}{r|rrrr}
        x+1 & x^3 & +3x^2 & -2x & +5 \\
        \hline
          & x^2 & +2x & -4 & \\
          & -(x^3+x^2) & & & \\
        \hline
          & & 2x^2 & -2x & +5 \\
          & -(2x^2 + 2x) & & & \\
        \hline
          & & & -4x & +5 \\
          & -(-4x - 4) & & & \\
        \hline
          & & & & 9 \\
    \end{array}
    \]

The quotient is \( x^2 + 2x - 4 \) and the remainder is \( 9 \). Thus:

\[
\frac{x^3 + 3x^2 - 2x + 5}{x + 1} = x^2 + 2x - 4 + \frac{9}{x+1}
\]

Would you like more details or have any questions?

Here are some questions you might want to explore:
1. How do you perform synthetic division?
2. What is the Remainder Theorem?
3. Can you explain the Factor Theorem?
4. How do you factor polynomials completely?
5. What are some common mistakes to avoid in polynomial division?

**Tip:** When performing polynomial long division, carefully align each step to ensure accuracy, and always check your work by multiplying the divisor by the quotient and adding the remainder to see if you get the original polynomial.

Learn how to divide the polynomial x^3 + 3x^2 - 2x + 5 by x + 1 using long division. Step-by-step solution with detailed explanations and examples. Suitable for high school students.

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