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=2
x equals 2

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

x=0
, x=1

Answer 1

To factor the polynomial \( 2x^5 - 4x^3 + 2x \), we can start by factoring out the common term from each part of the polynomial.

First, notice that each term has a factor of \(2x\):

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

Next, we need to factor \( x^4 - 2x^2 + 1 \). We can use the substitution \( y = x^2 \):

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

Thus, we can write:

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

Putting it all together, we have:

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

Now, we can solve the equation \( 2x^5 - 4x^3 + 2x = 0 \) by setting the factored equation to zero:

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

This gives us the following solutions:

1. \( 2x = 0 \) → \( x = 0 \)
2. \( (x - 1)^2 = 0 \) → \( x = 1 \) (with multiplicity 2)
3. \( (x + 1)^2 = 0 \) → \( x = -1 \) (with multiplicity 2)

Thus, the roots of the polynomial equation are:

- \( x = 0 \)
- \( x = 1 \)
- \( x = -1 \)

From the given options, the correct response is:

**x=−1, x=0, x=1**