Question

What is the result of using synthetic division to divide 3x4−5x3+4x2−7x+1
by x+1
?(1 point)
Responses

3x3−8x2+12x−19+20x+1
3 x cubed minus 8 x squared plus 12 x minus 19 plus the fraction with numerator 20 and denominator x plus 1

3x3−8x2+12x−8
3 x cubed minus 8 x squared plus 12 x minus 8

3x3−8x2+12x−8−7x+1
3 x cubed minus 8 x squared plus 12 x minus 8 minus the fraction with numerator 7 and denominator x plus 1

3x3−2x2+2x−5−4x+1

Answer 1

To use synthetic division to divide \(3x^4 - 5x^3 + 4x^2 - 7x + 1\) by \(x + 1\), we will substitute \(-1\) for \(x\) in synthetic division (since we set \(x + 1 = 0\) giving \(x = -1\)).

Here are the steps:

1. Write down the coefficients of the polynomial:
\(3, -5, 4, -7, 1\)

2. Set up synthetic division using \(-1\):

```
-1 | 3 -5 4 -7 1
| -3 8 -12 19
----------------
3 -8 12 -19 20
```

3. Bring down the first coefficient \(3\).

4. Multiply \(-1\) by \(3\) and write the result underneath \(-5\):
\(-1 \times 3 = -3\).

5. Add \(-5\) and \(-3\) to get \(-8\).

6. Repeat the process:
- Multiply \(-1\) by \(-8\) to get \(8\), and add to \(4\): \(4 + 8 = 12\).
- Multiply \(-1\) by \(12\) to get \(-12\), and add to \(-7\): \(-7 - 12 = -19\).
- Multiply \(-1\) by \(-19\) to get \(19\), and add to \(1\): \(1 + 19 = 20\).

7. The result of the synthetic division is therefore:
- Quotient: \(3x^3 - 8x^2 + 12x - 19\)
- Remainder: \(20\)

So, we can write the result as:
\[3x^3 - 8x^2 + 12x - 19 + \frac{20}{x + 1}\]

Thus, the correct response is:
**3x^3−8x^2+12x−19+20/(x+1)**.