Question

use the long division method to find the result when 2x^3+7x^2+19x+8 is divided by 2x+1

dont get another wrong answer

Answer 1

To divide the polynomial \(2x^3 + 7x^2 + 19x + 8\) by \(2x + 1\) using the long division method, we can follow these steps:

1. **Set up the long division**: Write the divisor \(2x + 1\) outside the division symbol and the dividend \(2x^3 + 7x^2 + 19x + 8\) inside.

```
______________________
2x + 1 | 2x^3 + 7x^2 + 19x + 8
```

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

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

4. **Subtract**: Write this result under the dividend and subtract.
```
x^2
______________________
2x + 1 | 2x^3 + 7x^2 + 19x + 8
-(2x^3 + x^2)
__________________
6x^2 + 19x + 8
```

5. **Bring down the next term**: The next term down is unchanged, so we now work with \(6x^2 + 19x + 8\).

6. **Repeat the process**: Divide the leading term \(6x^2\) by \(2x\):
\[
\frac{6x^2}{2x} = 3x
\]

7. **Multiply**: Multiply \(3x\) by the entire divisor:
\[
3x(2x + 1) = 6x^2 + 3x
\]

8. **Subtract again**:
```
x^2 + 3x
______________________
2x + 1 | 2x^3 + 7x^2 + 19x + 8
-(2x^3 + x^2)
__________________
6x^2 + 19x + 8
-(6x^2 + 3x)
__________________
16x + 8
```

9. **Continue the process**: Now take \(16x\) and divide by \(2x\):
\[
\frac{16x}{2x} = 8
\]

10. **Multiply**: Multiply \(8\) by the divisor:
\[
8(2x + 1) = 16x + 8
\]

11. **Final subtraction**:
```
x^2 + 3x + 8
______________________
2x + 1 | 2x^3 + 7x^2 + 19x + 8
-(2x^3 + x^2)
__________________
6x^2 + 19x + 8
-(6x^2 + 3x)
__________________
16x + 8
-(16x + 8)
__________________
0
```

12. **Conclusion**: The remainder is \(0\), which means the polynomial division is exact.

Thus, the result of dividing \(2x^3 + 7x^2 + 19x + 8\) by \(2x + 1\) is:
\[
\boxed{x^2 + 3x + 8}
\]