factor the high-degree polynomial 2x^5 -4x^3+2x to solve the polynomial equation 2x^5-4x^3+2x=0

Answer 1

To factor the polynomial equation $2x^5-4x^3+2x=0$, we start by factoring out the common terms.

1. Factor out the greatest common factor (GCF): The GCF of the terms $2x^5$, $-4x^3$, and $2x$ is $2x$. We factor this out:

   $$2x(x^4-2x^2+1)=0$$

2. Set each factor to zero: Now we can set each factor equal to zero to find the solutions.

   From $2x=0$: $$x=0$$

   Now we solve the quartic equation $x^4-2x^2+1=0$.

3. Substitute $y=x^2$: This substitution simplifies $x^4-2x^2+1=0$ to a quadratic form: $$y^2-2y+1=0$$

4. Factor the quadratic: The quadratic can be factored as: $$(y-1{)}^2=0$$

   Therefore, we have: $$y-1=0⟹y=1$$

5. Back-substitute for $x$: Remembering that $y=x^2$, we have: $$x^2=1$$

   Thus, solving for $x$ gives us: $$x=1\text{or}x=-1$$

6. Collect the solutions: Combining all the solutions we found: $$x=0,x=1,x=-1$$

Therefore, the full set of solutions to the equation $2x^5-4x^3+2x=0$ is: $$\boxed{{\displaystyle 0,1,-1}}$$