To determine which of the students is correct, we need to perform polynomial long division of the polynomial \( P(x) = x^4 − 3x^3 − 19x^2 + 3x + 18 \) by the factor \( x - 1 \).
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Performing Long Division of \( P(x) \) by \( x - 1 \):
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Divide the leading term of \( P(x) \) (which is \( x^4 \)) by the leading term of \( x - 1 \) (which is \( x \)): \[ x^4 ÷ x = x^3 \]
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Multiply \( x^3 \) by \( x - 1 \): \[ x^3(x - 1) = x^4 - x^3 \]
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Subtract this from \( P(x) \): \[ (x^4 - 3x^3 - 19x^2 + 3x + 18) - (x^4 - x^3) = -2x^3 - 19x^2 + 3x + 18 \]
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Now, take the leading term of the new polynomial \( -2x^3 \) and divide by \( x \): \[ -2x^3 ÷ x = -2x^2 \]
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Multiply \(-2x^2\) by \( x - 1 \): \[ -2x^2(x - 1) = -2x^3 + 2x^2 \]
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Subtract this from \(-2x^3 - 19x^2 + 3x + 18\): \[ (-2x^3 - 19x^2 + 3x + 18) - (-2x^3 + 2x^2) = -21x^2 + 3x + 18 \]
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Now, take \(-21x^2\) and divide by \(x\): \[ -21x^2 ÷ x = -21x \]
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Multiply \(-21x\) by \(x - 1\): \[ -21x(x - 1) = -21x^2 + 21x \]
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Subtract: \[ (-21x^2 + 3x + 18) - (-21x^2 + 21x) = -18x + 18 \]
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Now take \(-18x\) and divide by \(x\): \[ -18x ÷ x = -18 \]
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Multiply \(-18\) by \(x - 1\): \[ -18(x - 1) = -18x + 18 \]
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Subtract: \[ (-18x + 18) - (-18x + 18) = 0 \]
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After performing the division, we find that:
\[ P(x) = (x - 1)(x^3 - 2x^2 - 21x - 18) \]
- Identify the correct student based on the results:
- Student #1: \( x^2 + 9x + 18 \) - Incorrect.
- Student #2: \( x^3 − 2x^2 − 21x − 18 \) - Correct.
- Student #3: \( x^3 − 2x^2 − 21x + 18 \) - Incorrect.
The correct student is Student #2.
3 answers