Question

Find the solutions of the equation x4 − 2x3 − x2 + 2x = 0. (Enter your answers as a comma-separated list.)

Answer 1

Answer:

$S=\left \{ 0,-1,1,2 \right \}$

Step-by-step explanation:

There are several ways to solve this quartic equation. But since the coefficients, they repeat a=1,b=2,c=1,d=2, but e=0, and they are multiple of each other, then it is more convenient to work with factoring as the method of solving it.

As if it was a quadratic one.

$x^{4}-2x^{3}- x^{2} + 2x = 0\\x(x^{3}-2x^{2}-x+2)=0 \:Factoring \:out\\x[(x^{3}-2x^{2})+(-x+2)]=0 \:Grouping\\x[\mathbf{x^{2}}(x-2)+\mathbf{-1}(x+2)]=0 \:Rewriting\:the\:first\:factor\\x(x^{2}-1)(x-2)\:Expanding \:the \:first \:factor\\x(x-1)(x+1)(x-2)=0\\x=0,x=1,x=-1,x=2\\S=\left \{ 0,-1,1,2 \right \}$

Answer 2

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