Question

factor the high-degree polynomial 2x^5 -4x^3+2x to solve the polynomial equation 2x^5-4x^3+2x=0

Answer 1

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

1. **Factor out the greatest common factor (GCF):**
The GCF of the terms \(2x^5\), \(-4x^3\), and \(2x\) is \(2x\). We factor this out:

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

2. **Set each factor to zero:**
Now we can set each factor equal to zero to find the solutions.

From \(2x = 0\):
\[
x = 0
\]

Now we solve the quartic equation \(x^4 - 2x^2 + 1 = 0\).

3. **Substitute \(y = x^2\):**
This substitution simplifies \(x^4 - 2x^2 + 1 = 0\) to a quadratic form:
\[
y^2 - 2y + 1 = 0
\]

4. **Factor the quadratic:**
The quadratic can be factored as:
\[
(y - 1)^2 = 0
\]

Therefore, we have:
\[
y - 1 = 0 \implies y = 1
\]

5. **Back-substitute for \(x\):**
Remembering that \(y = x^2\), we have:
\[
x^2 = 1
\]

Thus, solving for \(x\) gives us:
\[
x = 1 \quad \text{or} \quad x = -1
\]

6. **Collect the solutions:**
Combining all the solutions we found:
\[
x = 0, \quad x = 1, \quad x = -1
\]

Therefore, the full set of solutions to the equation \(2x^5 - 4x^3 + 2x = 0\) is:
\[
\boxed{0, 1, -1}
\]