To solve the polynomial equation \( 2x^5 - 4x^3 + 2x = 0 \), we can start by factoring the polynomial.
-
Factor out the greatest common factor from the polynomial: \[ 2x^5 - 4x^3 + 2x = 2x(x^4 - 2x^2 + 1) \]
-
Next, we can simplify the polynomial inside the parentheses: \[ x^4 - 2x^2 + 1 \] This can be seen as a quadratic in terms of \( x^2 \). We can rewrite it as: \[ (x^2 - 1)^2 \]
-
So, we can further factor it: \[ (x^2 - 1)(x^2 - 1) = (x - 1)(x + 1)(x - 1)(x + 1) = (x - 1)^2(x + 1)^2 \]
-
Now, we can write our factored equation: \[ 2x(x - 1)^2(x + 1)^2 = 0 \]
-
Setting each factor to zero gives us: \[ 2x = 0 \quad \Rightarrow \quad x = 0 \] \[ (x - 1)^2 = 0 \quad \Rightarrow \quad x = 1 \quad (\text{double root}) \] \[ (x + 1)^2 = 0 \quad \Rightarrow \quad x = -1 \quad (\text{double root}) \]
Thus, the solutions to the equation \( 2x^5 - 4x^3 + 2x = 0 \) are:
- \( x = 0 \)
- \( x = 1 \) (with multiplicity 2)
- \( x = -1 \) (with multiplicity 2)
From the provided response options, the correct choice is: x=−1, x=0, x=1 (x equals negative 1, x equals 0, x equals 1).