Factor the high-degree polynomial 2x5−4x3+2x to solve the polynomial equation 2x5−4x3+2x=0.(1 point) Responses x=2 x equals 2 x=−1, x=0, x=1 x equals negative 1 , x equals 0 , x equals 1 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 first factor out the common term:

1. Factor out the greatest common factor:

   \[ 2x^5 - 4x^3 + 2x = 2x(x^4 - 2x^2 + 1) \]

2. Now we have:

   \[ 2x(x^4 - 2x^2 + 1) = 0 \]

   This gives us one solution: \( x = 0 \).

3. Next, we need to factor the quadratic-like expression \( x^4 - 2x^2 + 1 \). We can substitute \( y = x^2 \):

   \[ y^2 - 2y + 1 = (y - 1)^2 \]

   Substituting back \( y = x^2 \):

   \[ (x^2 - 1)^2 = (x - 1)^2(x + 1)^2 \]

4. Combine all the factors:

   \[ 2x(x - 1)^2(x + 1)^2 = 0 \]

5. Find the solutions from the factors:

   • From \( 2x = 0 \), we get \( x = 0 \).
   • From \( (x - 1)^2 = 0 \), we get \( x = 1 \) (with a multiplicity of 2).
   • From \( (x + 1)^2 = 0 \), we get \( x = -1 \) (with a multiplicity of 2).

6. Thus, the solutions to the polynomial equation are:

   \[ x = 0, \quad x = 1, \quad x = -1 \]

Final Answer:

The roots are \( x = 0, x = 1, x = -1 \).

Therefore, the correct response is: x = -1, x = 0, x = 1.