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 (1 point)
To find a factor of a polynomial, we can use polynomial long division. We divide the given polynomial, x^4−3x^3−19x^2+3x+18, by the linear factor x−1.
Student #1: x^2+9x+18
Using polynomial long division, we find:
x^2 - 2x - 16
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- x^4 + x^3
------------
- 4x^3 - 19x^2 + 3x
+ 4x^3 - 4x^2
------------
- 23x^2 + 3x + 18
+ 23x^2 - 23x
--------------
26x + 18
The remainder is not equal to zero, so x−1 is not a factor of x^4−3x^3−19x^2+3x+18. Hence, Student #1 is incorrect.
Student #2: x^3−2x^2−21x−18
Using polynomial long division, we find:
x^3 - 3x^2 - 16x + 15
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- x^4 + x^3
------------
- 4x^3 - 19x^2 + 3x
+ 4x^3 - 4x^2
------------
- 23x^2 + 3x + 18
+ 23x^2 - 23x
--------------
- 20x + 18
+ 20x - 20
------------
- 2
The remainder is not equal to zero, so x−1 is not a factor of x^4−3x^3−19x^2+3x+18. Hence, Student #2 is incorrect.
Student #3: x^3−2x^2−21x+18
Using polynomial long division, we find:
x^3 - 3x^2 - 16x + 15
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- x^4 + x^3
------------
- 4x^3 - 19x^2 + 3x
+ 4x^3 - 4x^2
------------
- 23x^2 + 3x + 18
+ 23x^2 - 23x
--------------
- 20x + 18
+ 20x - 20
------------
- 2
The remainder is not equal to zero, so x−1 is not a factor of x^4−3x^3−19x^2+3x+18. Hence, Student #3 is incorrect.
Therefore, none of the students are correct.