factor the high-degree polynomial 2x^5 -4x^3+2x to solve the polynomial equation 2x^5-4x^3+2x=0

Answer 1

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

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

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

2. Set each factor to zero: Now we can set each factor equal to zero to find the solutions.

   From \(2x = 0\): \[ x = 0 \]

   Now we solve the quartic equation \(x^4 - 2x^2 + 1 = 0\).

3. Substitute \(y = x^2\): This substitution simplifies \(x^4 - 2x^2 + 1 = 0\) to a quadratic form: \[ y^2 - 2y + 1 = 0 \]

4. Factor the quadratic: The quadratic can be factored as: \[ (y - 1)^2 = 0 \]

   Therefore, we have: \[ y - 1 = 0 \implies y = 1 \]

5. Back-substitute for \(x\): Remembering that \(y = x^2\), we have: \[ x^2 = 1 \]

   Thus, solving for \(x\) gives us: \[ x = 1 \quad \text{or} \quad x = -1 \]

6. Collect the solutions: Combining all the solutions we found: \[ x = 0, \quad x = 1, \quad x = -1 \]

Therefore, the full set of solutions to the equation \(2x^5 - 4x^3 + 2x = 0\) is: \[ \boxed{0, 1, -1} \]