To perform synthetic division on the polynomial \( x^4 - 2x^3 + x^2 + 4x - 3 \) by the divisor \( x - 1 \), we first write down the coefficients of the polynomial:
- Coefficients: \( 1, -2, 1, 4, -3 \)
Since we are dividing by \( x - 1 \), we use \( 1 \) (the root of \( x - 1 = 0 \)) for synthetic division.
The steps for synthetic division are as follows:
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Write down the coefficients: \[ 1 \quad -2 \quad 1 \quad 4 \quad -3 \]
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Set \( 1 \) to the left: \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & & & & \ \hline & & & & & \ \end{array} \]
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Bring down the leading coefficient (1): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & & & & \ \hline & 1 & & & & \ \end{array} \]
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Multiply \( 1 \) by the number just written below the line:
- \( 1 \times 1 = 1 \). Write this under the second coefficient. \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & & & \ \hline & 1 & & & & \ \end{array} \]
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Add the second column:
- \(-2 + 1 = -1\) \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & & & \ \hline & 1 & -1 & & & \ \end{array} \]
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Repeat this process:
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Multiply: \( 1 \times -1 = -1 \) and add to the next coefficient \( 1 + (-1) = 0\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & -1 & & \ \hline & 1 & -1 & 0 & & \ \end{array} \]
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Multiply: \( 1 \times 0 = 0 \) and add to \( 4 + 0 = 4\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & -1 & 0 & \ \hline & 1 & -1 & 0 & 4 & \ \end{array} \]
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Multiply: \( 1 \times 4 = 4 \) and add to \(-3 + 4 = 1\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & -1 & 0 & 4 \ \hline & 1 & -1 & 0 & 4 & 1 \ \end{array} \]
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The result of this synthetic division is:
- The quotient: \( x^3 - x^2 + 0x + 4\) or simply \( x^3 - x^2 + 4 \)
- The remainder: \( 1 \)
So the final result of the division is: \[ x^4 - 2x^3 + x^2 + 4x - 3 = (x - 1)(x^3 - x^2 + 4) + 1 \]