Solve 10x^5+25x^3+35x^2

$$10 x ^ { 5 } + 25 x ^ { 3 } + 35 x ^ { 2 }$$

Factor

$$5\left(x+1\right)x^{2}\left(2x^{2}-2x+7\right)$$

Solution Steps

Factor out 5.

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

Consider 2x^{5}+5x^{3}+7x^{2}. Factor out x^{2}.

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

Consider 2x^{3}+5x+7. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 7 and q divides the leading coefficient 2. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(2x^{2}-2x+7\right)$$

Rewrite the complete factored expression. Polynomial 2x^{2}-2x+7 is not factored since it does not have any rational roots.

$$5x^{2}\left(x+1\right)\left(2x^{2}-2x+7\right)$$

Evaluate

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