using the rational root theorem to list all possible rational roots of the polynomial equation
x^3-x^2-x-3=0
possible answers
-3,-1,1,3
1,3
-33
no roots
Please someone explain
To use the Rational Root Theorem to find all possible rational roots of a polynomial equation, you need to identify the factors of the constant term (in this case, -3) and the factors of the highest-degree coefficient (in this case, 1).
The factors of -3 are ±1 and ±3, and the factors of 1 are ±1.
These potential rational roots can be found by taking all possible combinations of these factors.
The possible rational roots for the given polynomial equation x^3 - x^2 - x - 3 = 0 are:
-3, -1, 1, and 3.
However, we can verify which of these roots are actually valid by substituting them one by one into the equation and checking if they make the equation true.
For x = -3, the equation becomes:
(-3)^3 - (-3)^2 - (-3) - 3 = 0
-27 - 9 + 3 - 3 = 0
-36 = 0
Since -36 does not equal 0, -3 is not a root.
For x = -1, the equation becomes:
(-1)^3 - (-1)^2 - (-1) - 3 = 0
-1 - 1 + 1 - 3 = 0
-4 = 0
Since -4 does not equal 0, -1 is not a root.
For x = 1, the equation becomes:
(1)^3 - (1)^2 - (1) - 3 = 0
1 - 1 - 1 - 3 = 0
-4 = 0
Since -4 does not equal 0, 1 is not a root.
For x = 3, the equation becomes:
(3)^3 - (3)^2 - (3) - 3 = 0
27 - 9 - 3 - 3 = 0
12 = 0
Since 12 does not equal 0, 3 is not a root either.
Therefore, the polynomial equation x^3 - x^2 - x - 3 = 0 has no rational roots.