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

Answer 1

To factor the polynomial equation \( 2x^5 - 4x^3 + 2x = 0 \), we can start by factoring out the common term:

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

   So we factor out \( 2x \): \[ 2x(x^4 - 2x^2 + 1) = 0 \]

2. Set each factor to zero: From \( 2x = 0 \), we find: \[ x = 0 \]

   Now, we need to solve the quartic equation \( x^4 - 2x^2 + 1 = 0 \). We can let \( y = x^2 \): \[ y^2 - 2y + 1 = 0 \]

   This is a perfect square: \[ (y - 1)^2 = 0 \]

   Thus, solving for \( y \): \[ y - 1 = 0 \implies y = 1 \] Re-substituting \( y = x^2 \): \[ x^2 = 1 \implies x = 1 \quad \text{or} \quad x = -1 \]

3. Summarize the solutions: The solutions to the original polynomial equation are: \[ x = 0, \quad x = 1, \quad x = -1 \]

Thus, the complete factorization 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.