Question

Factor the high-degree polynomial 2x5−4x3+2x
2
𝑥
5
−
4
𝑥
3
+
2
𝑥
to solve the polynomial equation 2x5−4x3+2x=0
2
𝑥
5
−
4
𝑥
3
+
2
𝑥
=
0
.(1 point)
Responses

x=0
𝑥
=
0
, x=1
𝑥
=
1
x equals 0 , x equals 1

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

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

x=2

Answer 1

To factor the polynomial \( 2x^5 - 4x^3 + 2x = 0 \), let's first factor out the greatest common factor (GCF):

1. Identify the GCF of the terms \( 2x^5 \), \( -4x^3 \), and \( 2x \). The GCF is \( 2x \).
2. Factor out \( 2x \) from the polynomial:

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

Next, we need to solve the equation:

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

This gives us one solution immediately:

\[
x = 0
\]

Now, we must solve the quartic polynomial \( x^4 - 2x^2 + 1 = 0 \). To make this easier, let’s use a substitution:

Let \( y = x^2 \). Then the equation becomes:

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

This factors to:

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

So we find:

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

Substituting back \( y = x^2 \):

\[
x^2 = 1 \implies x = \pm 1
\]

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

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

So, the correct response is:

**x equals -1, x equals 0, x equals 1.**