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)

Answer 1

To determine if a factor is correct, we can use polynomial long division.

Let's perform polynomial division using Student #1's factor:

x^3 + 9x^2 + 18x
_____________________________________
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- (x^4 - x^3) + 2x^2 - 4x - 18
_____________________________________
- 4x^3 - 17x^2 + 7x
+ ( 4x^3 - 4x^2) + 4x
_____________________________________
- 13x^2 + 11x + 18
+ (13x^2 - 13x) - 13
_____________________________________
24x + 5

As we can see, the factor x - 1 does not divide evenly into the polynomial x^4 - 3x^3 - 19x^2 + 3x + 18. Therefore, Student #1 is not correct.

Now, let's perform polynomial division using Student #2's factor:

x^2 + 2x + 18
_____________________________________
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- (x^4 - x^3) + 2x^2 - 2x - 18
_____________________________________
- 21x^2 + 5x
+ (21x^2 - 21x) + 21
_____________________________________
26x + 21

Again, the factor x - 1 does not divide evenly into the polynomial x^4 - 3x^3 - 19x^2 + 3x + 18. Therefore, Student #2 is not correct.

Finally, let's perform polynomial division using Student #3's factor:

x^2 + 21x + 18
_____________________________________
x - 1 | x^4 - 3x^3 - 19x^2 + 3x + 18
- (x^4 - x^3) + 2x^2 - 2x - 18
_____________________________________
- 21x^2 + 5x
+ (21x^2 - 21x) - 21
_____________________________________
4x - 3

Again, the factor x - 1 does not divide evenly into the polynomial x^4 - 3x^3 - 19x^2 + 3x + 18. Therefore, Student #3 is not correct.

None of the three students is correct in finding a factor of x^4 - 3x^3 - 19x^2 + 3x + 18, given that x - 1 is one of the linear factors.