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

Answer 1

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 \( x - 1 \).

1. **Performing Long Division of \( P(x) \) by \( x - 1 \)**:

- Divide the leading term of \( P(x) \) (which is \( x^4 \)) by the leading term of \( x - 1 \) (which is \( x \)):
\[
x^4 ÷ x = x^3
\]
- Multiply \( x^3 \) by \( x - 1 \):
\[
x^3(x - 1) = x^4 - x^3
\]
- Subtract this from \( P(x) \):
\[
(x^4 - 3x^3 - 19x^2 + 3x + 18) - (x^4 - x^3) = -2x^3 - 19x^2 + 3x + 18
\]

- Now, take the leading term of the new polynomial \( -2x^3 \) and divide by \( x \):
\[
-2x^3 ÷ x = -2x^2
\]
- Multiply \(-2x^2\) by \( x - 1 \):
\[
-2x^2(x - 1) = -2x^3 + 2x^2
\]
- Subtract this from \(-2x^3 - 19x^2 + 3x + 18\):
\[
(-2x^3 - 19x^2 + 3x + 18) - (-2x^3 + 2x^2) = -21x^2 + 3x + 18
\]

- Now, take \(-21x^2\) and divide by \(x\):
\[
-21x^2 ÷ x = -21x
\]
- Multiply \(-21x\) by \(x - 1\):
\[
-21x(x - 1) = -21x^2 + 21x
\]
- Subtract:
\[
(-21x^2 + 3x + 18) - (-21x^2 + 21x) = -18x + 18
\]

- Now take \(-18x\) and divide by \(x\):
\[
-18x ÷ x = -18
\]
- Multiply \(-18\) by \(x - 1\):
\[
-18(x - 1) = -18x + 18
\]
- Subtract:
\[
(-18x + 18) - (-18x + 18) = 0
\]

After performing the division, we find that:

\[
P(x) = (x - 1)(x^3 - 2x^2 - 21x - 18)
\]

2. **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**.