Question

Divide using long division. Check your answer.
(x^3 +3x^2 +x+6)/(x-1)

The quotient is ____ with a remainder of ____.
(Simplify your answers. Do not factor.)

Answer 1

To divide \(x^3 + 3x^2 + x + 6\) by \(x - 1\) using long division, follow these steps:

1. **Setup the Long Division**:
\[
\begin{array}{r|l}
& x^2 + 4x + 5 \\
\hline
x - 1 & x^3 + 3x^2 + x + 6 \\
\end{array}
\]

2. **Divide the leading terms**:
- Divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x\), which gives us \(x^2\).
- Write this \(x^2\) above the long division bar.

3. **Multiply**:
- Multiply \(x^2\) by the divisor \(x - 1\):
\[
x^2(x - 1) = x^3 - x^2
\]
- Write this under the dividend and subtract:
\[
(x^3 + 3x^2 + x + 6) - (x^3 - x^2) = (3x^2 + x^2) + x + 6 = 4x^2 + x + 6
\]

4. **Repeat**:
- Divide \(4x^2\) by \(x\) to get \(4x\).
- Add \(4x\) above the bar.
- Multiply \(4x\) by \(x - 1\):
\[
4x(x - 1) = 4x^2 - 4x
\]
- Subtract:
\[
(4x^2 + x + 6) - (4x^2 - 4x) = (x + 4x) + 6 = 5x + 6
\]

5. **Continue**:
- Divide \(5x\) by \(x\) to get \(5\).
- Add \(5\) above the bar.
- Multiply \(5\) by \(x - 1\):
\[
5(x - 1) = 5x - 5
\]
- Subtract:
\[
(5x + 6) - (5x - 5) = 6 + 5 = 11
\]

6. **Final Result**:
- We end up with \(11\) as the remainder.

The long division process yields:
\[
x^2 + 4x + 5 \quad \text{(quotient)}
\]
with a remainder of:
\[
11
\]

So, the answer is:
- The quotient is \(x^2 + 4x + 5\) with a remainder of \(11\).