Question

Mr. Gonzalez asks his top three students to find a factor of x4−3x3−19x2+3x+18
, given that x−1
is one of the linear factors. Which of the students is correct?

Student #1: x2+9x+18


Student #2: x3−2x2−21x−18


Student #3: x3−2x2−21x+18


(1 point)
Student #
is correct.

Answer 1

To determine which student 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 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 this from the original polynomial:
\[
(x^4 - 3x^3) - (x^4 - x^3) = -2x^3.
\]
- Bring down the next term, resulting in:
\[
-2x^3 - 19x^2.
\]

- Divide \( -2x^3 \) by \( x \) to get \( -2x^2 \).
- Multiply \( -2x^2 \) by \( x - 1 \) to get \( -2x^3 + 2x^2 \).
- Subtract:
\[
(-2x^3 - 19x^2) - (-2x^3 + 2x^2) = -21x^2.
\]
- Bring down the next term, resulting in:
\[
-21x^2 + 3x.
\]

- Divide \( -21x^2 \) by \( x \) to get \( -21x \).
- Multiply by \( x - 1 \) to get \( -21x^2 + 21x \).
- Subtract:
\[
(-21x^2 + 3x) - (-21x^2 + 21x) = -18x.
\]
- Bring down the last term, resulting in:
\[
-18x + 18.
\]

- Divide \( -18x \) by \( x \) to get \( -18 \).
- Multiply by \( x - 1 \) to get \( -18x + 18 \).
- Subtract:
\[
(-18x + 18) - (-18x + 18) = 0.
\]

Thus, the polynomial division yields:
\[
x^4 - 3x^3 - 19x^2 + 3x + 18 = (x - 1)(x^3 - 2x^2 - 21x - 18).
\]

2. **Identify the correct factor**:
The quotient we obtained from the division is \( x^3 - 2x^2 - 21x - 18 \).

Now, we will check which student's answer matches this quotient:

- **Student #1**: \( x^2 + 9x + 18 \) (not correct)
- **Student #2**: \( x^3 - 2x^2 - 21x - 18 \) (correct)
- **Student #3**: \( x^3 - 2x^2 - 21x + 18 \) (not correct)

Thus, **Student #2 is correct**.