Solve 6x^6-21x^5+15x^4

$$6 x ^ { 6 } - 21 x ^ { 5 } + 15 x ^ { 4 }$$

Factor

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

Solution Steps

Factor out 3.

$$3\left(2x^{6}-7x^{5}+5x^{4}\right)$$

Consider 2x^{6}-7x^{5}+5x^{4}. Factor out x^{4}.

$$x^{4}\left(2x^{2}-7x+5\right)$$

Consider 2x^{2}-7x+5. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-7 ab=2\times 5=10$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 10.

$$-1,-10 -2,-5$$

Calculate the sum for each pair.

$$-1-10=-11 -2-5=-7$$

The solution is the pair that gives sum -7.

$$a=-5 b=-2$$

Rewrite 2x^{2}-7x+5 as \left(2x^{2}-5x\right)+\left(-2x+5\right).

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

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

$$x\left(2x-5\right)-\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(x-1\right)$$

Rewrite the complete factored expression.

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

Evaluate

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