To find out which of the students 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 Polynomial Long 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 \( x^4 - x^3 \) from \( x^4 - 3x^3 - 19x^2 + 3x + 18 \) to get: \[ (-3x^3 + x^3) - 19x^2 + 3x + 18 = -2x^3 - 19x^2 + 3x + 18 \]
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Divide the leading term \( -2x^3 \) by \( x \), which gives \( -2x^2 \).
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Multiply \( -2x^2 \) by \( x - 1 \) to get \( -2x^3 + 2x^2 \).
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Subtract \( -2x^3 + 2x^2 \) from the previous polynomial: \[ (-2x^3 + 2x^2) - 19x^2 + 3x + 18 = -21x^2 + 3x + 18 \]
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Divide the leading term \( -21x^2 \) by \( x \), which gives \( -21x \).
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Multiply \( -21x \) by \( x - 1 \) to get \( -21x^2 + 21x \).
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Subtract \( -21x^2 + 21x \) from the previous polynomial: \[ (-21x^2 + 21x) + 3x + 18 = -18x + 18 \]
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Finally, divide the leading term \( -18x \) by \( x \), which gives \( -18 \).
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Multiply \( -18 \) by \( x - 1 \) to get \( -18x + 18 \).
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Subtract \( -18x + 18 \) from the previous polynomial: \[ (-18x + 18) - (-18x + 18) = 0 \]
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This gives us the result of the division: \[ x^4 - 3x^3 - 19x^2 + 3x + 18 = (x - 1)(x^3 - 2x^2 - 21x - 18). \]
- Identify the Correct Student:
- From the polynomial division, we can see that the other factor is \( x^3 - 2x^2 - 21x - 18 \).
- This matches the answer given by Student #2.
Therefore, the correct answer is:
B - Student #2