) Use long division to solve: (3𝑥

Question

3
+ 5𝑥
2 − 2𝑥) ÷ (𝑥 + 2)

Answer 1

To perform long division on the polynomial \( (3x^3 + 5x^2 - 2x) \) divided by \( (x + 2) \), we'll follow the standard long division process.

1. Set up the long division:

          _____________________
x + 2 | 3x^3 + 5x^2 - 2x

2. Divide the first term:

• The leading term of the dividend is \( 3x^3 \) and the leading term of the divisor is \( x \).
• Divide \( 3x^3 \) by \( x \): \[ 3x^2 \]
• Write it above the long division line.

3. Multiply and subtract:

• Multiply \( 3x^2 \) by \( (x + 2) \): \[ 3x^2 \cdot (x + 2) = 3x^3 + 6x^2 \]
• Now, subtract this from the original polynomial: \[ 3x^3 + 5x^2 - 2x - (3x^3 + 6x^2) = (5x^2 - 6x^2) - 2x = -x^2 - 2x \]

4. Bring down the next term:

• We now treat \(-x^2 - 2x\) as our new dividend.

          3x^2
          _____________________
x + 2 | 3x^3 + 5x^2 - 2x
          -(3x^3 + 6x^2)
          ______________________
                     -x^2 - 2x

5. Repeat the process:

• Divide \(-x^2\) by \(x\): \[ -x \]
• Write it above the line.

6. Multiply and subtract:

• Multiply \(-x\) by \((x + 2)\): \[ -x \cdot (x + 2) = -x^2 - 2x \]
• Subtract this from \(-x^2 - 2x\): \[ -x^2 - 2x - (-x^2 - 2x) = 0 \]

7. Bring down the next term:

• The remainder is 0, and we have no more terms to bring down.

          3x^2 - x
          _____________________
x + 2 | 3x^3 + 5x^2 - 2x
          -(3x^3 + 6x^2)
          ______________________
                     -x^2 - 2x
                     -(-x^2 - 2x)
          ______________________
                              0

8. Final result: Since the remainder is 0, the complete factorization is: \[ \frac{3x^3 + 5x^2 - 2x}{x + 2} = 3x^2 - x \]

So, the final answer is: \[ 3x^2 - x \]