To solve the polynomial division \( \frac{x^4 - 2x^3 + x^2 + 4x - 3}{x - 1} \) using synthetic division, we will follow these steps:
- Use the root \( r = 1 \) for the divisor \( x - 1 \).
- Write down the coefficients of the polynomial: \( 1, -2, 1, 4, -3 \).
- Set up synthetic division.
Here's how the synthetic division is performed step-by-step:
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Write the coefficients: \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & & & & \ \end{array} \]
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Bring down the first coefficient (1): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & & & & \ & 1 & & & & \ \end{array} \]
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Multiply \( 1 \) by \( 1 \) and add to the next coefficient \(-2\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & & & \ & 1 & -1 & & & \ \end{array} \]
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Continue the process:
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Multiply \( 1 \) by \(-1\) and add to \(1\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & -1 & & \ & 1 & -1 & 0 & & \ \end{array} \]
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Multiply \( 1 \) by \(0\) and add to \(4\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & -1 & 0 & \ & 1 & -1 & 0 & 4 & \ \end{array} \]
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Multiply \( 1 \) by \(4\) and add to \(-3\): \[ \begin{array}{r|rrrrr} 1 & 1 & -2 & 1 & 4 & -3 \ & & 1 & -1 & 0 & 4 \ & 1 & -1 & 0 & 4 & 1 \ \end{array} \]
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The bottom row gives us the quotient and the remainder. The result will be:
\[ 1x^3 - 1x^2 + 0x + 4 \text{ with a remainder of } 1. \]
Thus, the answer in the format of polynomial division is:
\[ x^3 - x^2 + 4 + \frac{1}{x-1}. \]
From the options you provided, this corresponds to the entry:
"The bottom row has entries of 1, negative 1, 0, 4, and 1."
Therefore, the correct response is:
"A synthetic division display with a divisor of 1. The first row has entries of 1, negative 2, 1, 4, and negative 3. The second row has entries of 1, negative 1, 0, and 4. The bottom row has entries of 1, negative 1, 0, 4, and 1."