Question

) Use long division to solve: (3𝑥
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:
\[
3x^2 - x
\]