Answer:
Therefore, the solutions to the equation x^3 - 6x^2 - 9x + 54 = 0 are:
x = 3, x = 6, and x = -3.
Explanation:
To solve the given equation x^3 - 6x^2 - 9x + 54 = 0, we can use the Rational Root Theorem followed by synthetic division or other factoring methods.
First, let's list all possible rational roots of the equation. The rational root theorem states that any rational roots of the equation are of the form p/q, where p is a factor of the constant term (in this case, 54) and q is a factor of the leading coefficient (in this case, 1).
The factors of 54 are ±1, ±2, ±3, ±6, ±9, ±18, ±27, ±54.
The factors of 1 are ±1.
By testing these factors in the equation, we find that x = 3 is a root of the given equation.
Now, we can use polynomial division or synthetic division to further factorize and solve the quadratic equation. Performing the division, we get:
(x^3 - 6x^2 - 9x + 54) ÷ (x - 3) = x^2 - 3x - 18.
Now, we can factor the quadratic equation x^2 - 3x - 18. Factoring, we get (x - 6)(x + 3) = 0.
Setting each factor equal to zero, we get:
x - 6 = 0 and x + 3 = 0.
Solving for x in each equation, we find:
x = 6 and x = -3.
Therefore, the solutions to the equation x^3 - 6x^2 - 9x + 54 = 0 are:
x = 3, x = 6, and x = -3.