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

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

GPT-4o mini · Answer

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**.