Mr. Gonzalez asks his top three students to find a factor of x4−3x3−19x2+3x+18

Question

, 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 if a polynomial is a factor of another polynomial, we can use synthetic division or polynomial division.

Let's check each student's answer one by one:

Student #1: x^2 + 9x + 18

Let's divide the given polynomial (x^4 - 3x^3 - 19x^2 + 3x + 18) by the supposed factor (x - 1):

x^3 10x^2 -8x -11
___________________________________
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- x^4 + x^3 + 19x^2 - 19x
___________________________________
x^3 + 10x^2 - 16x + 18
- x^3 + x^2 - x^2 + x
___________________________________
9x^2 - 15x + 18
- 9x^2 + 9x - 9
___________________________________
-6x + 27

The result of the division is not zero, which means that x - 1 is not a factor of the original polynomial. Therefore, student #1 is incorrect.

Student #2: x^3 - 2x^2 - 21x - 18

Let's divide the given polynomial (x^4 - 3x^3 - 19x^2 + 3x + 18) by the supposed factor (x - 1):

x^3 - 3x^2 - 16x - 5
___________________________________
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- x^4 x^3 19x^2 -19x
___________________________________
-4x^3 - 38x^2 + 22x + 18
4x^3 - 4x^2 4x - 4
___________________________________
-34x^2 + 26x + 14
34x^2 - 34x 34
___________________________________
-8x + 48

The result of the division is not zero, which means that x - 1 is not a factor of the original polynomial. Therefore, student #2 is incorrect.

Student #3: x^3 - 2x^2 - 21x + 18

Let's divide the given polynomial (x^4 - 3x^3 - 19x^2 + 3x + 18) by the supposed factor (x - 1):

x^3 - 4x^2 + x + 18
___________________________________
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- x^4 x^3 19x^2 -19x
___________________________________
-4x^3 - 22x^2 + 22x + 18
4x^3 - 4x^2 4x - 4
___________________________________
-26x^2 + 26x + 14
26x^2 - 26x 26
___________________________________
0

The result of the division is zero, which means that x - 1 is a factor of the original polynomial. Therefore, student #3 is correct.

In conclusion, student #3 is the only one who is correct.