Factor the high-degree polynomial 2x5−4x3+2x to solve the polynomial equation 2x5−4x3+2x=0.(1 point)

Question

Responses

x=−1, x=0, x=1
x equals negative 1 , x equals 0 , x equals 1

x=2
x equals 2

x=0, x=1
x equals 0 , x equals 1

x=−1, x=1

Answer 1

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

1. Factor out the GCF (Greatest Common Factor): $$2x^5-4x^3+2x=2x(x^4-2x^2+1)$$

2. Set the factored polynomial equal to zero: $$2x(x^4-2x^2+1)=0$$

   This gives us one solution immediately: $$2x=0\Rightarrow x=0$$

3. Now, we need to solve $x^4-2x^2+1=0$. Let's make a substitution $y=x^2$: $$y^2-2y+1=0$$

   This factors as: $$(y-1{)}^2=0$$

   Hence, we find: $$y-1=0\Rightarrow y=1$$

   Substituting back gives: $$x^2=1\Rightarrow x=1\text{or}x=-1$$

4. Collecting all the solutions: The potential solutions are: $$x=0,x=1,x=-1$$

Therefore, the final factored form of the polynomial and the solutions to the equation $2x^5-4x^3+2x=0$ are $x=0,x=1,x=-1$.

The correct response is: x = −1, x = 0, x = 1.