Answer:
x = 0, 1, 2 and -1
Step-by-step explanation:
x⁴ − 2x³ − x² + 2x = 0.
Factorizing the equation,
x(x³ − 2x² - x + 2) = 0
⇒ x = 0 and x³ − 2x² - x + 2 = 0
⇒ x = 0 is a root of the equation
Solving for the polynomial x³ - 2x² - x + 2 = 0
When x = 1
(1)³ - 2(1)² - 1 + 2
1 - 2 - 1 + 2 = 0
Therefore, x = 1 is a root of the equation.
⇒ x - 1 = 0
Using long division approach to get the other roots
x² - x - 2
-----------------------
x - 1 ║ x³ - 2x² - x + 2
x³ - x²
---------------------
- x² - x + 2
- x² + x
------------------------
- 2x + 2
- 2x + 2
-------------------
0
We will solve the quotient x² - x - 2 to get the other roots of the equation.
Solving quadratic equation x² - x - 2 = 0 using factorization method
x² - 2x + x - 2 = 0
(x² - 2x) + (x - 2) = 0
x(x - 2) + 1(x - 2) = 0
(x - 2)(x + 1) = 0
x - 2 = 0 and x + 1 = 0
⇒ x = 2 and x = - 1 are also roots of the equations.
The solution of the equation x⁴ − 2x³ − x² + 2x = 0 is x = 0, x = 1, x = 2 and x = -1