Solve the polynomial equation p(x)=x3+2x2−5x−6 . Which of the following is a factor?(1 point) Responses x−3 x minus 3 x+2 x plus 2 x−1 x minus 1 x+1
To check if a given factor is a factor of a polynomial, we can use synthetic division.
Let's check if x - 3 is a factor of p(x) = x^3 + 2x^2 - 5x - 6.
Using synthetic division:
3 | 1 2 -5 -6
|______ 3 15 30
_________________
1 5 10 24
The remainder is 24, not zero. Therefore, x - 3 is not a factor of p(x).
Now let's check x + 2 using synthetic division:
-2 | 1 2 -5 -6
|_____ -2 0 10
__________________
1 0 -5 4
The remainder is 4, not zero. Therefore, x + 2 is not a factor of p(x).
Now let's check x - 1 using synthetic division:
1 | 1 2 -5 -6
|______ 1 3 -2
_________________
1 3 -2 -8
The remainder is -8, not zero. Therefore, x - 1 is not a factor of p(x).
Lastly, let's check x + 1 using synthetic division:
-1 | 1 2 -5 -6
|_____ -1 -1 6
__________________
1 1 -6 0
The remainder is zero. Therefore, x + 1 is a factor of p(x).
Hence, the correct option is: x + 1.