Solve 2x^3-4x^4+2x^2

$$2 x ^ { 3 } - 4 x ^ { 4 } + 2 x ^ { 2 }$$

Factor

$$2\left(1-x\right)\left(2x+1\right)x^{2}$$

Solution Steps

Factor out 2.

$$2\left(x^{3}-2x^{4}+x^{2}\right)$$

Consider x^{3}-2x^{4}+x^{2}. Factor out x^{2}.

$$x^{2}\left(x-2x^{2}+1\right)$$

Consider x-2x^{2}+1. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-2x^{2}+x+1$$

Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=1 ab=-2=-2$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=2 b=-1$$

Rewrite -2x^{2}+x+1 as \left(-2x^{2}+2x\right)+\left(-x+1\right).

$$\left(-2x^{2}+2x\right)+\left(-x+1\right)$$

Factor out 2x in -2x^{2}+2x.

$$2x\left(-x+1\right)-x+1$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(2x+1\right)$$

Rewrite the complete factored expression.

$$2x^{2}\left(-x+1\right)\left(2x+1\right)$$

Evaluate

$$2\left(1-x\right)\left(2x+1\right)x^{2}$$