Question

Use long division to divide the polynomials: x3−x2+4/x+2

Answer 1

To divide the polynomial \( x^3 - x^2 + 4 \) by \( x + 2 \) using long division, follow these steps:

1. **Setup the long division**: Write \( x^3 - x^2 + 0x + 4 \) (adding \( 0x \) for the missing term) under the long division symbol, and write \( x + 2 \) outside.

```
____________
x + 2 | x^3 - x^2 + 0x + 4
```

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

3. **Multiply and subtract**: Multiply \( x^2 \) by \( x + 2 \) and subtract it from the original polynomial:
\[
x^2(x + 2) = x^3 + 2x^2
\]
Subtract this from the original polynomial:
\[
(x^3 - x^2) - (x^3 + 2x^2) = -x^2 - 2x^2 = -3x^2
\]
Now, bring down the next term, which is \( 0x \):
\[
-3x^2 + 0x = -3x^2
\]

We write:
```
x^2
____________
x + 2 | x^3 - x^2 + 0x + 4
- (x^3 + 2x^2)
__________
-3x^2 + 0x
```

4. **Repeat with -3x^2**: Now, divide the leading term \(-3x^2\) by the leading term of the divisor:
\[
\frac{-3x^2}{x} = -3x
\]

5. **Multiply and subtract again**:
\[
-3x(x + 2) = -3x^2 - 6x
\]
Subtract this from the current remainder:
\[
(-3x^2 + 0x) - (-3x^2 - 6x) = 0x + 6x = 6x
\]

Now, we have:
```
x^2 - 3x
____________
x + 2 | x^3 - x^2 + 0x + 4
- (x^3 + 2x^2)
__________
-3x^2 + 0x
- (-3x^2 - 6x)
__________
6x
```

6. **Repeat with 6x**: Now, divide \( 6x \) by the leading term of the divisor:
\[
\frac{6x}{x} = 6
\]

7. **Multiply and subtract one last time**:
\[
6(x + 2) = 6x + 12
\]
Subtract this from the current polynomial:
\[
(6x + 4) - (6x + 12) = 4 - 12 = -8
\]

Putting this all together:
```
x^2 - 3x + 6
____________
x + 2 | x^3 - x^2 + 0x + 4
- (x^3 + 2x^2)
__________
-3x^2 + 0x
- (-3x^2 - 6x)
__________
6x + 4
- (6x + 12)
__________
-8
```

8. **Final answer**: The quotient is \( x^2 - 3x + 6 \) and the remainder is \( -8 \). Thus, we can express the final result of the division as:
\[
\frac{x^3 - x^2 + 4}{x + 2} = x^2 - 3x + 6 - \frac{8}{x + 2}
\]

So, the answer is:
\[
\boxed{x^2 - 3x + 6 - \frac{8}{x + 2}}
\]