Solve 8x^5+12x^3+20x^2

$$8 x ^ { 5 } + 12 x ^ { 3 } + 20 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 4.

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

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

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

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

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

Rewrite the complete factored expression. Polynomial 2x^{2}-2x+5 is not factored since it does not have any rational roots.

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

Evaluate

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