To divide the polynomial \(x^4 - 3x^3 + 6x^2 - 12x + 8\) by \(x - 1\) using synthetic division, we follow these steps:
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Set up synthetic division: Since we are dividing by \(x - 1\), we will use \(1\) (the root) for synthetic division.
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Write down the coefficients: The coefficients of the polynomial are \(1\), \(-3\), \(6\), \(-12\), and \(8\).
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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 \(8 - 8 = 0\).
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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:
\[ \boxed{x^3 - 2x^2 + 4x - 8} \]