To evaluate \( (3x^2 - 6x + 3) \div (x - 1) \) using synthetic division, we'll follow these steps:
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Identify the coefficients of the polynomial \( 3x^2 - 6x + 3 \). The coefficients are:
- \( 3 \) (for \( x^2 \))
- \( -6 \) (for \( x^1 \))
- \( 3 \) (for the constant term)
So we have the coefficients: \( [3, -6, 3] \).
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Use the root of the divisor \( x - 1 \), which is \( 1 \).
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Set up the synthetic division:
\[ \begin{array}{r|rrr} 1 & 3 & -6 & 3 \ & & 3 & -3 \ \hline & 3 & -3 & 0 \ \end{array} \]
Here's how we perform the synthetic division:
- Bring down the \( 3 \).
- Multiply \( 1 \) (the divisor root) by \( 3 \) (the number we just brought down) to get \( 3 \), and add it to \(-6\) to get \(-3\).
- Multiply \( 1 \) by \(-3\) to get \(-3\), and add it to \( 3 \) to get \( 0 \).
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Interpret the result: The bottom row shows the coefficients of the quotient polynomial, which is \( 3x - 3 \), and the final value \( 0 \) indicates that \( (x - 1) \) is a factor of \( 3x^2 - 6x + 3 \).
Thus, the result of \( (3x^2 - 6x + 3) \div (x - 1) \) is
\[ \boxed{3x - 3} \]