Question

Use the long division method to find the result when 2, x, cubed, plus, 7, x, squared, plus, 19, x, plus, 82x
3
+7x
2
+19x+8 is divided by 2, x, plus, 12x+1.



Solve the problem just like the example below
⬇️
Use the long division method to find the result when
4, x, cubed, plus, 5, x, squared, plus, 5, x, plus, 1
4x
3
+5x
2
+5x+1 is divided by
4, x, plus, 1
4x+1.
Use Long Division:
Use Long Division:


x, squared
x
2
plus, x
+x
plus, 1
+1


4, x
4x
plus, 1
+1
right parenthesis
)
4, x, cubed
4x
3
plus, 5, x, squared
+5x
2
plus, 5, x
+5x
plus, 1
+1



start superscript, minus, end superscript, 4, x, cubed
−
4x
3
plus, start superscript, minus, end superscript, x, squared
+
−
x
2






4, x, squared
4x
2
plus, 5, x
+5x
plus, 1
+1




start superscript, minus, end superscript, 4, x, squared
−
4x
2
plus, start superscript, minus, end superscript, x
+
−
x






4, x
4x
plus, 1
+1





start superscript, minus, end superscript, 4, x
−
4x
plus, start superscript, minus, end superscript, 1
+
−
1






0
0

Answer: , x, squared, plus, x, plus, 1
Answer: x
2
+x+1

Answer 1

To divide \(2x^3 + 7x^2 + 19x + 82\) by \(2x + 12\) using long division, we will follow the method step-by-step.

### Long Division

1. **Setup the Long Division**:
- Divide \(2x^3 + 7x^2 + 19x + 82\) by \(2x + 12\).

```
________________
2x + 12 | 2x^3 + 7x^2 + 19x + 82
```

2. **Divide the leading terms**:
- Divide \(2x^3\) by \(2x\) to get \(x^2\).

```
x^2
________________
2x + 12 | 2x^3 + 7x^2 + 19x + 82
```

3. **Multiply and subtract**:
- Multiply \(x^2\) by \(2x + 12\) to get \(2x^3 + 12x^2\).
- Subtract this from \(2x^3 + 7x^2\):

```
x^2
________________
2x + 12 | 2x^3 + 7x^2 + 19x + 82
-(2x^3 + 12x^2)
________________
-5x^2 + 19x
```

4. **Bring down the next term**:
- Bring down the \(+82\) to get \(-5x^2 + 19x + 82\).

```
x^2
________________
2x + 12 | 2x^3 + 7x^2 + 19x + 82
-(2x^3 + 12x^2)
________________
-5x^2 + 19x + 82
```

5. **Divide the leading terms again**:
- Divide \(-5x^2\) by \(2x\) to get \(-\frac{5}{2}x\).

```
x^2 - (5/2)x
________________
2x + 12 | 2x^3 + 7x^2 + 19x + 82
-(2x^3 + 12x^2)
________________
-5x^2 + 19x + 82
-(-5x^2 - 30x)
______________________
49x + 82
```

6. **Multiply and subtract**:
- Multiply \(-\frac{5}{2}x\) by \(2x + 12\) to get \(-5x^2 - 30x\).
- Subtract this from the previous remainder:

```
49x + 82
- (-5x^2 - 30x)
______________________
49x + 82
- (49x + 588)
_______________________
-506
```

7. **Divide the constant**:
- Divide \(49x\) by \(2x\) to get \(24.5\), but for simplicity, we will keep it in the form we've got.

8. **Final Result**:
- The final step gives us the remainder which is \(-506\).
- Hence, we can write the answer:

```
Answer: x^2 - (5/2)x + 24.5 with a remainder of -506
```

Combining everything, you can represent the answer as:

**Answer**: \(x^2 - \frac{5}{2}x + 24.5 - \frac{506}{2x + 12}\)