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
, x=1
x equals 0 , x equals 1

Answer 1

To factor the polynomial \(2x^5 - 4x^3 + 2x\), we first look for common factors.

1. Notice that all terms share a common factor of \(2x\).

2. We can factor out \(2x\):

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

3. Now let's simplify the inner polynomial \(x^4 - 2x^2 + 1\). We can substitute \(y = x^2\), transforming it into a quadratic:

\[
y^2 - 2y + 1 = (y - 1)^2 = (x^2 - 1)^2
\]

4. So, we have:

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

5. Thus, we can further factor the original polynomial:

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

Now, we set the polynomial equal to zero to find the roots:

\[
2x(x - 1)^2(x + 1)^2 = 0
\]

This gives us the solutions:

- \(2x = 0 \Rightarrow x = 0\)
- \((x - 1)^2 = 0 \Rightarrow x = 1\) (with multiplicity 2)
- \((x + 1)^2 = 0 \Rightarrow x = -1\) (with multiplicity 2)

The solutions to the polynomial equation \(2x^5 - 4x^3 + 2x = 0\) are:

\[
x = -1, \quad x = 0, \quad x = 1
\]

Thus, the correct response is:

**x = -1, x = 0, x = 1**