Mr. Gonzalez asks his top three students to find a factor of x

4
−3x
3
−19x
2
+3x+18, given that x−1 is one of the linear factors. Which of the students is correct?
Student #1: x
2
+9x+18
Student #2: x
3
−2x
2
−21x−18
Student #3: x
3
−2x
2
−21x+18
A. Student #1 B. Student #2 C. Student #3 D. None of the students

1 answer

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 \( x - 1 \).

  1. Divide the leading term: \( x^4 \div x = x^3 \).
  2. Multiply \( x^3 \) by \( (x - 1) \): \[ x^3(x - 1) = x^4 - x^3 \]
  3. Subtract: \[ (x^4 - 3x^3) - (x^4 - x^3) = -2x^3 \]
  4. Bring down the next term: \[ -2x^3 - 19x^2 \]
  5. Divide the leading term: \( -2x^3 \div x = -2x^2 \).
  6. Multiply \( -2x^2 \) by \( (x - 1) \): \[ -2x^2(x - 1) = -2x^3 + 2x^2 \]
  7. Subtract: \[ (-2x^3 - 19x^2) - (-2x^3 + 2x^2) = -21x^2 \]
  8. Bring down the next term: \[ -21x^2 + 3x \]
  9. Divide the leading term: \( -21x^2 \div x = -21x \).
  10. Multiply \( -21x \) by \( (x - 1) \): \[ -21x(x - 1) = -21x^2 + 21x \]
  11. Subtract: \[ (-21x^2 + 3x) - (-21x^2 + 21x) = -18x \]
  12. Bring down the next term: \[ -18x + 18 \]
  13. Divide the leading term: \( -18x \div x = -18 \).
  14. Multiply \( -18 \) by \( (x - 1) \): \[ -18(x - 1) = -18x + 18 \]
  15. 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 \( x^3 - 2x^2 - 21x - 18 \)