To determine which student is correct, we can use polynomial long division to divide the polynomial \( x^4 - 3x^3 - 19x^2 + 3x + 18 \) by the factor \( x - 1 \).
- Perform the division:
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Divide the leading term \( x^4 \) by \( x \), which gives \( x^3 \).
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Multiply \( x^3 \) by \( x - 1 \) to get \( x^4 - x^3 \).
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Subtract this from the original polynomial: \[ (x^4 - 3x^3) - (x^4 - x^3) = -2x^3. \]
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Bring down the next term, resulting in: \[ -2x^3 - 19x^2. \]
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Divide \( -2x^3 \) by \( x \) to get \( -2x^2 \).
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Multiply \( -2x^2 \) by \( x - 1 \) to get \( -2x^3 + 2x^2 \).
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Subtract: \[ (-2x^3 - 19x^2) - (-2x^3 + 2x^2) = -21x^2. \]
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Bring down the next term, resulting in: \[ -21x^2 + 3x. \]
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Divide \( -21x^2 \) by \( x \) to get \( -21x \).
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Multiply by \( x - 1 \) to get \( -21x^2 + 21x \).
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Subtract: \[ (-21x^2 + 3x) - (-21x^2 + 21x) = -18x. \]
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Bring down the last term, resulting in: \[ -18x + 18. \]
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Divide \( -18x \) by \( x \) to get \( -18 \).
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Multiply by \( x - 1 \) to get \( -18x + 18 \).
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Subtract: \[ (-18x + 18) - (-18x + 18) = 0. \]
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Thus, the polynomial division yields: \[ x^4 - 3x^3 - 19x^2 + 3x + 18 = (x - 1)(x^3 - 2x^2 - 21x - 18). \]
- Identify the correct factor: The quotient we obtained from the division is \( x^3 - 2x^2 - 21x - 18 \).
Now, we will check which student's answer matches this quotient:
- Student #1: \( x^2 + 9x + 18 \) (not correct)
- Student #2: \( x^3 - 2x^2 - 21x - 18 \) (correct)
- Student #3: \( x^3 - 2x^2 - 21x + 18 \) (not correct)
Thus, Student #2 is correct.