Solve 2x^6-12x^5+10x^4

$$2 { x }^{ 6 } -12 { x }^{ 5 } +10 { x }^{ 4 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(x^{6}-6x^{5}+5x^{4}\right)$$

Consider x^{6}-6x^{5}+5x^{4}. Factor out x^{4}.

$$x^{4}\left(x^{2}-6x+5\right)$$

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.

$$a+b=-6 ab=1\times 5=5$$

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.

$$a=-5 b=-1$$

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 the first and -1 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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