Question

Factor the high-degree polynomial 2x5−4x3+2x to solve the polynomial equation 2x5−4x3+2x=0.(1 point)
Responses

x=−1, x=1
x equals negative 1 , x equals 1

x=−1, x=0, x=1
x equals negative 1 , x equals 0 , x equals 1

x=2
x equals 2

x=0,

Answer 1

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

1. Factor out the greatest common factor from the polynomial:
\[
2x^5 - 4x^3 + 2x = 2x(x^4 - 2x^2 + 1)
\]

2. 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:
\[
(x^2 - 1)^2
\]

3. So, we can further factor it:
\[
(x^2 - 1)(x^2 - 1) = (x - 1)(x + 1)(x - 1)(x + 1) = (x - 1)^2(x + 1)^2
\]

4. Now, we can write our factored equation:
\[
2x(x - 1)^2(x + 1)^2 = 0
\]

5. 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).