Question

Divide using synthetic division. Write your answer in standard form.



(
x
4
−
3
x
3
+
6
x
2
−
12
x
+
8
)
÷
(
x
−
1
)
(x
4
−3x
3
+6x
2
−12x+8)÷(x−1)

Answer 1

To divide the polynomial \(x^4 - 3x^3 + 6x^2 - 12x + 8\) by \(x - 1\) using synthetic division, we follow these steps:

1. **Set up synthetic division:** Since we are dividing by \(x - 1\), we will use \(1\) (the root) for synthetic division.

2. **Write down the coefficients:** The coefficients of the polynomial are \(1\), \(-3\), \(6\), \(-12\), and \(8\).

3. **Perform synthetic division:**

\[
\begin{array}{r|rrrrr}
1 & 1 & -3 & 6 & -12 & 8 \\
& & 1 & -2 & 4 & -8 \\
\hline
& 1 & -2 & 4 & -8 & 0 \\
\end{array}
\]

- Bring down the \(1\).
- Multiply \(1\) (the divisor) by \(1\) (the current coefficient) and write it under the next coefficient: \(1 \times 1 = 1\).
- Add \(-3 + 1 = -2\).
- Multiply \(1\) by \(-2\) and write it under the next coefficient: \(1 \times -2 = -2\).
- Add \(6 - 2 = 4\).
- Multiply \(1\) by \(4\): \(1 \times 4 = 4\).
- Add \(-12 + 4 = -8\).
- Multiply \(1\) by \(-8\): \(1 \times -8 = -8\).
- Add \(8 - 8 = 0\).

4. **Write the result:** The coefficients \(1\), \(-2\), \(4\), and \(-8\) represent the polynomial of degree one less than the original polynomial:

\[
x^3 - 2x^2 + 4x - 8
\]

Since the remainder is \(0\), there is no additional term.

Thus, the result of the division \((x^4 - 3x^3 + 6x^2 - 12x + 8) \div (x - 1)\) in standard form is:

\[
\boxed{x^3 - 2x^2 + 4x - 8}
\]