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

4. Collecting all the solutions: The potential solutions are: \[ x = 0, \quad x = 1, \quad x = -1 \]

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.