X^3-5x+4=0

To solve the equation x^3 - 5x + 4 = 0, we can use any of the several methods. One method is to use the Rational Root Theorem, which helps us identify possible rational roots of the equation.

The Rational Root Theorem states that if a polynomial equation has a rational root, it must be in the form p/q, where p is a factor of the constant term (4 in this case) and q is a factor of the leading coefficient (1 in this case).

In our equation, the possible rational roots can be found by considering factors of 4, such as ±1, ±2, and ±4. We can then substitute each of these values of x into the equation to see if we get a zero.

Let's start by checking the possible rational roots:

For x = 1: (1)^3 - 5(1) + 4 = 1 - 5 + 4 = 0 (satisfied) For x = -1: (-1)^3 - 5(-1) + 4 = -1 + 5 + 4 = 8 (not satisfied) For x = 2: (2)^3 - 5(2) + 4 = 8 - 10 + 4 = 2 (not satisfied) For x = -2: (-2)^3 - 5(-2) + 4 = -8 + 10 + 4 = 6 (not satisfied) For x = 4: (4)^3 - 5(4) + 4 = 64 - 20 + 4 = 48 (not satisfied) For x = -4: (-4)^3 - 5(-4) + 4 = -64 + 20 + 4 = -40 (not satisfied)

From this analysis, we find that x = 1 is a root of the equation, which means (x - 1) is a factor of the equation.

To solve the equation further, we can use polynomial long division or synthetic division with (x - 1) as the divisor.

By performing the division, we get: x^3 - 5x + 4 = (x - 1)(x^2 + x - 4)

Now, we have factored the equation as (x - 1)(x^2 + x - 4) = 0.

To solve further, we can set each factor equal to zero and solve for x:

Setting x - 1 = 0, we get x = 1.

Setting x^2 + x - 4 = 0, we can use the quadratic formula: x = (-1 ± √(1^2 - 4(1)(-4))) / (2(1)) x = (-1 ± √(1 + 16)) / 2 x = (-1 ± √17) / 2

Therefore, the solutions to the equation x^3 - 5x + 4 = 0 are: x = 1, x = (-1 + √17) / 2, and x = (-1 - √17) / 2.