Solve 8v^6+2v^5-10v^9

$$8 v ^ { 6 } + 2 v ^ { 5 } - 10 v ^ { 9 }$$

Factor

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

Solution Steps

Factor out 2.

$$2\left(4v^{6}+v^{5}-5v^{9}\right)$$

Consider 4v^{6}+v^{5}-5v^{9}. Factor out v^{5}.

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

Consider 4v+1-5v^{4}. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient -5. One such root is 1. Factor the polynomial by dividing it by v-1.

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

Rewrite the complete factored expression. Polynomial -5v^{3}-5v^{2}-5v-1 is not factored since it does not have any rational roots.

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

Evaluate

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