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

Question

A.x=-1 x=1
B. X=-1 x=0 x=1
C. X=2
D. X=0 x=1

Answer 1

To solve the polynomial equation \( 2x^5 - 4x^3 + 2x = 0 \), we can start by factoring the expression.

1. Factor out the greatest common factor (GCF): The GCF of the terms in the polynomial is \( 2x \). So, we can factor that out:

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

2. Set the factored terms to zero:

   This gives us two equations to solve for zeros: \[ 2x = 0 \quad \text{and} \quad x^4 - 2x^2 + 1 = 0 \]

   From \( 2x = 0 \), we find: \[ x = 0 \]

3. Next, we solve the quartic polynomial: Letting \( y = x^2 \), we rewrite the quartic as: \[ y^2 - 2y + 1 = 0 \] This can be factored as: \[ (y - 1)^2 = 0 \] Thus, we have: \[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \] Reverting back to \( x \): \[ x^2 = 1 \quad \Rightarrow \quad x = \pm 1 \]

4. Sum up all the solutions: The complete set of solutions to the equation \( 2x^5 - 4x^3 + 2x = 0 \) is: \[ x = 0, \quad x = 1, \quad x = -1 \]

5. Conclusion: The correct set of roots we found is \( x = -1, x = 0, x = 1 \).

Hence, the answer is: B. \( x = -1, x = 0, x = 1 \).