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