To divide the polynomial \( x^3 - x^2 + 4 \) by \( x + 2 \) using long division, follow these steps:
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Set up the long division: \[ \begin{array}{r|rrrr} & 1 & -1 & 0 & 4 \ x+2 & x^3 & -x^2 & 0 & 4 \ \end{array} \]
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Divide the leading term: Divide the leading term \( x^3 \) by the leading term of the divisor \( x \): \[ \frac{x^3}{x} = x^2 \] Place \( x^2 \) above the line.
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Multiply: Multiply \( x^2 \) by \( x + 2 \) and write it underneath: \[ x^2(x + 2) = x^3 + 2x^2 \] \[ \begin{array}{r|rrrr} & 1 & -1 & 0 & 4 \ x+2 & x^3 & -x^2 & 0 & 4 \ & & x^3 & +2x^2 & \ \hline & & 0 & -3x^2 & 4 \ \end{array} \]
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Subtract: Subtract the result from the previous row: \[ 0 - (-3x^2) + 0 + 4 = -3x^2 + 4 \]
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Repeat the process: Now, divide the leading term \( -3x^2 \) by the leading term \( x \): \[ \frac{-3x^2}{x} = -3x \] Place \( -3x \) above the line.
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Multiply: Multiply \( -3x \) by \( x + 2 \): \[ -3x(x + 2) = -3x^2 - 6x \] Write it beneath \(-3x^2 + 4\): \[ \begin{array}{r|rrrr} & 1 & -1 & 0 & 4 \ x+2 & x^3 & -x^2 & 0 & 4 \ & & x^3 & +2x^2 & \ & & 0 & -3x^2 & 4 \ & & & -3x^2 & -6x \ \hline & & & 0 & 10x + 4 \ \end{array} \]
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Subtract: Subtract the result from the previous row: \[ 4 - (-6x) = 10x + 4 \]
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Repeat again: Divide \( 10x \) by \( x \): \[ \frac{10x}{x} = 10 \] Place \( 10 \) above the line.
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Multiply: Multiply \( 10 \) by \( x + 2 \): \[ 10(x + 2) = 10x + 20 \] Write it beneath \( 10x + 4 \): \[ \begin{array}{r|rrrr} & 1 & -1 & 0 & 4 \ x+2 & x^3 & -x^2 & 0 & 4 \ & & x^3 & +2x^2 & \ & & 0 & -3x^2 & 4 \ & & & -3x^2 & -6x \ & & & 0 & 10x + 4 \ & & & & 10x + 20 \ \hline & & & & 0 & -16 \ \end{array} \]
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Subtract: Subtract the results: \[ 4 - 20 = -16 \]
So the quotient is: \[ \text{Quotient} = x^2 - 3x + 10 \] and the remainder is: \[ \text{Remainder} = -16 \]
Thus, the result of the polynomial long division of \( \frac{x^3 - x^2 + 4}{x + 2} \) is: \[ x^2 - 3x + 10 - \frac{16}{x + 2} \]
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