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

Question

A.x=-1 x=1
B. X=-1 x=0 x=1
C. X=2
D. X=0 x=1

Answer 1

To solve the polynomial equation $2x^5-4x^3+2x=0$, we can start by factoring the expression.

1. Factor out the greatest common factor (GCF): The GCF of the terms in the polynomial is $2x$. So, we can factor that out:

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

2. Set the factored terms to zero:

   This gives us two equations to solve for zeros: $$2x=0\text{and}{x}^4-2x^2+1=0$$

   From $2x=0$, we find: $$x=0$$

3. Next, we solve the quartic polynomial: Letting $y=x^2$, we rewrite the quartic as: $$y^2-2y+1=0$$ This can be factored as: $$(y-1{)}^2=0$$ Thus, we have: $$y-1=0\Rightarrow y=1$$ Reverting back to $x$: $$x^2=1\Rightarrow x=\pm 1$$

4. Sum up all the solutions: The complete set of solutions to the equation $2x^5-4x^3+2x=0$ is: $$x=0,x=1,x=-1$$

5. Conclusion: The correct set of roots we found is $x=-1,x=0,x=1$.

Hence, the answer is: B. $x=-1,x=0,x=1$.