Question

factor the high-degree polynomial 2x^5 -4x^3+2x to solve the polynomial equation 2x^5-4x^3+2x=0
A.x=-1 x=1
B. X=-1 x=0 x=1
C. X=2
D. X=0 x=1

Answer 1

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

1. **Factor out the greatest common factor (GCF)**:
The GCF of the terms in the polynomial is \( 2x \). So, we can factor that out:

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

2. **Set the factored terms to zero**:

This gives us two equations to solve for zeros:
\[
2x = 0 \quad \text{and} \quad x^4 - 2x^2 + 1 = 0
\]

From \( 2x = 0 \), we find:
\[
x = 0
\]

3. **Next, we solve the quartic polynomial**:
Letting \( y = x^2 \), we rewrite the quartic as:
\[
y^2 - 2y + 1 = 0
\]
This can be factored as:
\[
(y - 1)^2 = 0
\]
Thus, we have:
\[
y - 1 = 0 \quad \Rightarrow \quad y = 1
\]
Reverting back to \( x \):
\[
x^2 = 1 \quad \Rightarrow \quad x = \pm 1
\]

4. **Sum up all the solutions**:
The complete set of solutions to the equation \( 2x^5 - 4x^3 + 2x = 0 \) is:
\[
x = 0, \quad x = 1, \quad x = -1
\]

5. **Conclusion**:
The correct set of roots we found is \( x = -1, x = 0, x = 1 \).

Hence, the answer is:
**B. \( x = -1, x = 0, x = 1 \)**.