Solve -x^4+6x^3-5x^2

$$- { x }^{ 4 } +6 { x }^{ 3 } -5 { x }^{ 2 }$$

Factor

$$\left(1-x\right)\left(x-5\right)x^{2}$$

Solution Steps

Factor out x^{2}.

$$x^{2}\left(-x^{2}+6x-5\right)$$

Consider -x^{2}+6x-5. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-5\right)=5$$

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

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

$$\left(-x^{2}+5x\right)+\left(x-5\right)$$

Factor out -x in -x^{2}+5x.

$$-x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+1\right)$$

Rewrite the complete factored expression.

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

Evaluate

$$\left(1-x\right)\left(x-5\right)x^{2}$$