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

Question

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 proceed as follows:

1. Factor out the common terms: The polynomial \( 2x^5 - 4x^3 + 2x \) has a common factor of \( 2x \):

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

2. Set the factored form to zero: We have two factors to consider:

   \[ 2x = 0 \quad \text{or} \quad x^4 - 2x^2 + 1 = 0 \]

   From \( 2x = 0 \), we get:

   \[ x = 0 \]

3. Solve the quartic equation: The quartic equation can be simplified by letting \( y = x^2 \):

   \[ y^2 - 2y + 1 = 0 \]

   This factors to:

   \[ (y - 1)^2 = 0 \implies y = 1 \]

   Since \( y = x^2 \), we then have:

   \[ x^2 = 1 \implies x = 1 \quad \text{or} \quad x = -1 \]

4. Final solutions: Combining all solutions, we have:

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

Thus, the solutions to the polynomial equation \( 2x^5 - 4x^3 + 2x = 0 \) are \( x = -1, x = 0, x = 1 \).

The correct response is:

x = −1, x = 0, x = 1.