Mr. Gonzalez asks his top three students to find a factor of x^4-3x^3-19x^2+3x+18

Question

, given that x-1
is one of the linear factors. Which of the students is correct?

Student #1:


Student #2:



Student #3:



A
Student #1

B
Student #2

C
Student #3

D
None of the students

Answer 1

To determine which student is correct in finding a factor of the polynomial \(x^4 - 3x^3 - 19x^2 + 3x + 18\) given that \(x - 1\) is one of the linear factors, we can use polynomial long division to divide the polynomial by \(x - 1\).

1. Synthetic Division / Long Division:

   Since \(x - 1\) is a factor, we perform synthetic division using \(x = 1\):

   For \(x^4 - 3x^3 - 19x^2 + 3x + 18\):

   • Coefficients: \(1, -3, -19, 3, 18\)

   Using synthetic division with \(1\):

   1 |  1   -3   -19    3    18
     |       1   -2   -21  -18
     ----------------------------
       1   -2   -21  -18   0
   

   The result after synthetic division gives us:

   \[ x^3 - 2x^2 - 21x - 18 \]

2. Finding Further Factors:

   Now that we have reduced the polynomial to \(x^3 - 2x^2 - 21x - 18\), we can look for factors of this cubic polynomial.

   Testing potential rational roots (factors of the constant term -18):

   • Testing \(x = 3\): \[ 3^3 - 2(3^2) - 21(3) - 18 = 27 - 18 - 63 - 18 = -72 \quad \text{(not a root)} \]

   • Testing \(x = -2\): \[ (-2)^3 - 2(-2)^2 - 21(-2) - 18 = -8 - 8 + 42 - 18 = 8 \quad \text{(not a root)} \]

   • Testing \(x = -3\): \[ (-3)^3 - 2(-3)^2 - 21(-3) - 18 = -27 - 18 + 63 - 18 = 0 \quad \text{(is a root)} \]

   Since \(-3\) is a root, \(x + 3\) is a factor. Now we divide \(x^3 - 2x^2 - 21x - 18\) by \(x + 3\):

   Performing synthetic division using \(-3\):

   -3 |  1   -2   -21   -18
      |      -3   15    18
      -----------------------
        1   -5   -6     0
   

   The result is: \[ x^2 - 5x - 6 \]

3. Factoring the Quadratic:

   We can factor \(x^2 - 5x - 6\): \[ x^2 - 5x - 6 = (x - 6)(x + 1) \]

4. The Complete Factorization:

   The polynomial \(x^4 - 3x^3 - 19x^2 + 3x + 18\) factors completely as: \[ (x - 1)(x + 3)(x - 6)(x + 1) \]

From all these steps, we see that the factors \(x - 1\), \(x + 3\), and the others were derived correctly.

Now to identify which student identified the factors correctly, we would need the specific claims made by each student. However, if none of their statements accurately reflect the findings we've derived, then:

The answer could be D: None of the students.

If you provide the statements from each student, I can check if any of them is indeed correct.