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

a. x=2
b. x=-1, x=1
c. x=-1, x=0, x=1
d. x=0, x=1

1 answer

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

  1. Factor out the common term: Notice that all terms in the polynomial have a common factor of \( 2x \).

\[ 2x^5 - 4x^3 + 2x = 2x(x^4 - 2x^2 + 1) \]

  1. Factor the quadratic: Now we need to factor \( x^4 - 2x^2 + 1 \). This can be recognized as a quadratic in terms of \( x^2 \). We can set \( y = x^2 \): \[ y^2 - 2y + 1 = (y - 1)^2 = (x^2 - 1)^2 \]

  2. Combine the factors: Therefore, \[ x^4 - 2x^2 + 1 = (x^2 - 1)^2 = (x - 1)^2(x + 1)^2 \]

  3. Complete factorization: Now, we combine all the factors: \[ 2x(x - 1)^2(x + 1)^2 = 0 \]

  4. Solve for roots: Setting each factor to zero gives us:

    • From \( 2x = 0 \): \( x = 0 \)
    • From \( (x - 1)^2 = 0 \): \( x = 1 \) (with multiplicity 2)
    • From \( (x + 1)^2 = 0 \): \( x = -1 \) (with multiplicity 2)

Thus, the roots of the polynomial equation are \( x = 0 \), \( x = 1 \), and \( x = -1 \).

In terms of the choices given, the correct solution is:

c. x = -1, x = 0, x = 1