Answer: The solutions to the equation x^3 - 4x + 3 = 0 are x = 1 and x = (-1 ± √13) / 2.
Explanation:
AI-generated answer
To find the values of x for which the rational function r(x) equals zero, we need to solve the equation r(x) = 0. In this case, the rational function is given by r(x) = (x^3 - 4x + 3) / (x^4 + 2x - 4).
To solve this equation, we set the numerator equal to zero and solve for x:
x^3 - 4x + 3 = 0
This equation can be solved using various methods, such as factoring, the rational root theorem, or numerical methods. In this case, let's use factoring to find the solutions.
The equation x^3 - 4x + 3 = 0 can be factored as (x - 1)(x^2 + x - 3) = 0.
Setting each factor equal to zero, we have:
x - 1 = 0 or x^2 + x - 3 = 0
Solving the first equation, we find x = 1.
For the second equation, we can use the quadratic formula to find the solutions:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 1, and c = -3. Plugging these values into the quadratic formula, we get:
x = (-1 ± √(1^2 - 4(1)(-3))) / (2(1))
x = (-1 ± √(1 + 12)) / 2
x = (-1 ± √13) / 2