To determine which student is correct regarding the factor of the polynomial \( x^4 - 3x^3 - 19x^2 + 3x + 18 \), given that \( x - 1 \) is a factor, we can use polynomial long division to divide the polynomial by \( x - 1 \).
Step 1: Polynomial Long Division
Let's perform the division of \( x^4 - 3x^3 - 19x^2 + 3x + 18 \) by \( x - 1 \).
- Divide the leading term: \( x^4 \div x = x^3 \).
- Multiply \( x^3 \) by \( (x - 1) \): \[ x^3(x - 1) = x^4 - x^3 \]
- Subtract: \[ (x^4 - 3x^3) - (x^4 - x^3) = -2x^3 \]
- Bring down the next term: \[ -2x^3 - 19x^2 \]
- Divide the leading term: \( -2x^3 \div x = -2x^2 \).
- Multiply \( -2x^2 \) by \( (x - 1) \): \[ -2x^2(x - 1) = -2x^3 + 2x^2 \]
- Subtract: \[ (-2x^3 - 19x^2) - (-2x^3 + 2x^2) = -21x^2 \]
- Bring down the next term: \[ -21x^2 + 3x \]
- Divide the leading term: \( -21x^2 \div x = -21x \).
- Multiply \( -21x \) by \( (x - 1) \): \[ -21x(x - 1) = -21x^2 + 21x \]
- Subtract: \[ (-21x^2 + 3x) - (-21x^2 + 21x) = -18x \]
- Bring down the next term: \[ -18x + 18 \]
- Divide the leading term: \( -18x \div x = -18 \).
- Multiply \( -18 \) by \( (x - 1) \): \[ -18(x - 1) = -18x + 18 \]
- Subtract: \[ (-18x + 18) - (-18x + 18) = 0 \]
Result of the Division
The result of the polynomial long division is: \[ x^3 - 2x^2 - 21x - 18 \]
Thus, we have: \[ x^4 - 3x^3 - 19x^2 + 3x + 18 = (x - 1)(x^3 - 2x^2 - 21x - 18) \]
Verifying Students' Answers
Now, let’s compare the result \( x^3 - 2x^2 - 21x - 18 \) to the factors provided by the students:
- 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)
Conclusion
The correct factor, given that \( x-1 \) is a linear factor, is provided by:
B. Student #2 \( x^3 - 2x^2 - 21x - 18 \)