Question

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

Answer 1

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

The GCF of the terms \(2x^5\), \(-4x^3\), and \(2x\) is \(2x\). So, we can factor the equation as:

\[
2x(x^4 - 2x^2 + 1) = 0
\]

Now we can set each factor equal to zero. The first factor gives us:

\[
2x = 0 \implies x = 0
\]

Next, we need to solve the quartic polynomial \(x^4 - 2x^2 + 1 = 0\). We can make a substitution to simplify this. Let \(y = x^2\). Then the equation becomes:

\[
y^2 - 2y + 1 = 0
\]

This is a perfect square trinomial, and it can be factored as:

\[
(y - 1)^2 = 0
\]

Setting this equal to zero gives us:

\[
y - 1 = 0 \implies y = 1
\]

Substituting back for \(x^2\):

\[
x^2 = 1 \implies x = 1 \quad \text{or} \quad x = -1
\]

Combining all the solutions, we have:

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

Thus, the complete set of solutions to the polynomial equation \(2x^5 - 4x^3 + 2x = 0\) is:

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

The correct response is:

**x = −1, x = 0, x = 1**.