Solve 4x^5-20x^4+12x^3-60x^2

$$4 x ^ { 5 } - 20 x ^ { 4 } + 12 x ^ { 3 } - 60 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 4.

$$4\left(x^{5}-5x^{4}+3x^{3}-15x^{2}\right)$$

Consider x^{5}-5x^{4}+3x^{3}-15x^{2}. Factor out x^{2}.

$$x^{2}\left(x^{3}-5x^{2}+3x-15\right)$$

Consider x^{3}-5x^{2}+3x-15. Do the grouping x^{3}-5x^{2}+3x-15=\left(x^{3}-5x^{2}\right)+\left(3x-15\right), and factor out x^{2} in the first and 3 in the second group.

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

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

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

Rewrite the complete factored expression. Polynomial x^{2}+3 is not factored since it does not have any rational roots.

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

Evaluate

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