Solve 4x^4-8x^3+5x^2-x

$$4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(4x^{3}-8x^{2}+5x-1\right)$$

Consider 4x^{3}-8x^{2}+5x-1. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 4. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1.

$$\left(2x-1\right)\left(2x^{2}-3x+1\right)$$

Consider 2x^{2}-3x+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.

$$a+b=-3 ab=2\times 1=2$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-2 b=-1$$

Rewrite 2x^{2}-3x+1 as \left(2x^{2}-2x\right)+\left(-x+1\right).

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

Factor out 2x in the first and -1 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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