Question

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

Answer 1

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 \( x - 1 \).

1. **Perform the Polynomial Long Division:**
- Divide the leading term \( x^4 \) by \( x \), which gives \( x^3 \).
- Multiply \( x^3 \) by \( x - 1 \) to get \( x^4 - x^3 \).
- Subtract \( x^4 - x^3 \) from \( x^4 - 3x^3 - 19x^2 + 3x + 18 \) to get:
\[
(-3x^3 + x^3) - 19x^2 + 3x + 18 = -2x^3 - 19x^2 + 3x + 18
\]

- Divide the leading term \( -2x^3 \) by \( x \), which gives \( -2x^2 \).
- Multiply \( -2x^2 \) by \( x - 1 \) to get \( -2x^3 + 2x^2 \).
- Subtract \( -2x^3 + 2x^2 \) from the previous polynomial:
\[
(-2x^3 + 2x^2) - 19x^2 + 3x + 18 = -21x^2 + 3x + 18
\]

- Divide the leading term \( -21x^2 \) by \( x \), which gives \( -21x \).
- Multiply \( -21x \) by \( x - 1 \) to get \( -21x^2 + 21x \).
- Subtract \( -21x^2 + 21x \) from the previous polynomial:
\[
(-21x^2 + 21x) + 3x + 18 = -18x + 18
\]

- Finally, divide the leading term \( -18x \) by \( x \), which gives \( -18 \).
- Multiply \( -18 \) by \( x - 1 \) to get \( -18x + 18 \).
- Subtract \( -18x + 18 \) from the previous polynomial:
\[
(-18x + 18) - (-18x + 18) = 0
\]

This gives us the result of the division:
\[
x^4 - 3x^3 - 19x^2 + 3x + 18 = (x - 1)(x^3 - 2x^2 - 21x - 18).
\]

2. **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