To solve the polynomial equation \(2x^5 - 4x^3 + 2x = 0\), we can start by factoring out the greatest common factor (GCF) from each term.
The GCF of the terms \(2x^5\), \(-4x^3\), and \(2x\) is \(2x\). So, we can factor the equation as:
\[ 2x(x^4 - 2x^2 + 1) = 0 \]
Now we can set each factor equal to zero. The first factor gives us:
\[ 2x = 0 \implies x = 0 \]
Next, we need to solve the quartic polynomial \(x^4 - 2x^2 + 1 = 0\). We can make a substitution to simplify this. Let \(y = x^2\). Then the equation becomes:
\[ y^2 - 2y + 1 = 0 \]
This is a perfect square trinomial, and it can be factored as:
\[ (y - 1)^2 = 0 \]
Setting this equal to zero gives us:
\[ y - 1 = 0 \implies y = 1 \]
Substituting back for \(x^2\):
\[ x^2 = 1 \implies x = 1 \quad \text{or} \quad x = -1 \]
Combining all the solutions, we have:
\[ x = 0, \quad x = 1, \quad x = -1 \]
Thus, the complete set of solutions to the polynomial equation \(2x^5 - 4x^3 + 2x = 0\) is:
\[ x = 0, \quad x = 1, \quad x = -1 \]
The correct response is:
x = −1, x = 0, x = 1.