X^3-5x+4=0

To solve the equation x^3 - 5x + 4 = 0, you can use the Rational Root Theorem or factor theorem to find a rational root if one exists, and then use synthetic division to solve the equation.

Using synthetic division method:

Step 1: Based on the leading coefficient and constant term, the potential rational roots of the equation are ±1, ±2, ±4.

Step 2: Test the potential roots using synthetic division.

For x = 1: 1 | 1 -5 0 4 | 1 -4 -4 ------------- 1 -4 -4 0

Since the remainder is 0, x = 1 is a root.

Step 3: Now we need to find the other roots. To do this, divide the result of the previous division (quotient: 1 - 4 - 4) by x - 1 (the factor you found above).

(x - 1) | 1 -4 -4 | 1 -3 ------------- 1 -3 -7

So, the equation can be factored as (x - 1)(x^2 - 3x - 7) = 0.

Step 4: Solve the quadratic equation x^2 - 3x - 7 = 0.

Using the quadratic formula: x = (-b ± √(b^2 - 4ac))/2a

a = 1 b = -3 c = -7

x = (-(-3) ± √((-3)^2 - 4(1)(-7)))/(2(1)) x = (3 ± √(9 + 28))/2 x = (3 ± √37)/2

So, the final solutions to the equation x^3 - 5x + 4 = 0 are: x = 1 x = (3 + √37)/2 x = (3 - √37)/2