Question

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

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

x=2
x equals 2

x=0, x=1
x equals 0 , x equals 1

x=−1, x=1

Answer 1

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

1. **Factor out the GCF (Greatest Common Factor)**:
\[
2x^5 - 4x^3 + 2x = 2x(x^4 - 2x^2 + 1)
\]

2. **Set the factored polynomial equal to zero**:
\[
2x(x^4 - 2x^2 + 1) = 0
\]

This gives us one solution immediately:
\[
2x = 0 \Rightarrow x = 0
\]

3. **Now, we need to solve \( x^4 - 2x^2 + 1 = 0 \)**. Let's make a substitution \( y = x^2 \):
\[
y^2 - 2y + 1 = 0
\]

This factors as:
\[
(y - 1)^2 = 0
\]

Hence, we find:
\[
y - 1 = 0 \Rightarrow y = 1
\]

Substituting back gives:
\[
x^2 = 1 \Rightarrow x = 1 \quad \text{or} \quad x = -1
\]

4. **Collecting all the solutions**:
The potential solutions are:
\[
x = 0, \quad x = 1, \quad x = -1
\]

Therefore, the final factored form of the polynomial and the solutions to the equation \( 2x^5 - 4x^3 + 2x = 0 \) are \( x = 0, x = 1, x = -1 \).

The correct response is:
**x = −1, x = 0, x = 1**.