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 = 0, x = 1
C. x = -1, x = 0, x = 1
D. x = 2

Answer 1

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

1. Factor out the greatest common factor (GCF), which is \( 2x \): \[ 2x(x^4 - 2x^2 + 1) = 0 \]

2. Now, we can set each factor equal to zero: \[ 2x = 0 \quad \Rightarrow \quad x = 0 \]

   Next, we need to solve the quartic polynomial \( x^4 - 2x^2 + 1 = 0 \). We can use substitution to make it easier. Let \( y = x^2 \). Then, the equation becomes: \[ y^2 - 2y + 1 = 0 \]

3. This is a perfect square: \[ (y - 1)^2 = 0 \] So, \( y - 1 = 0 \) implies \( y = 1 \).

4. Since \( y = x^2 \), we now have: \[ x^2 = 1 \quad \Rightarrow \quad x = 1 \quad \text{or} \quad x = -1 \]

5. The solutions to the original equation \( 2x^5 - 4x^3 + 2x = 0 \) are: \[ x = 0, \quad x = 1, \quad x = -1 \]

Thus, the correct factors of the polynomial and the corresponding solutions based on the provided options are:

Answer: C. \( x = -1, x = 0, x = 1 \).