Solve f(w)=8w^6+3w^5+5w

$$f ( w ) = 8 w ^ { 6 } + 3 w ^ { 5 } + 5 w$$

Factor

$$w\left(w+1\right)\left(8w^{4}-5w^{3}+5w^{2}-5w+5\right)$$

Solution Steps

Factor out w.

$$w\left(8w^{5}+3w^{4}+5\right)$$

Consider 8w^{5}+3w^{4}+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 8. One such root is -1. Factor the polynomial by dividing it by w+1.

$$\left(w+1\right)\left(8w^{4}-5w^{3}+5w^{2}-5w+5\right)$$

Rewrite the complete factored expression. Polynomial 8w^{4}-5w^{3}+5w^{2}-5w+5 is not factored since it does not have any rational roots.

$$w\left(w+1\right)\left(8w^{4}-5w^{3}+5w^{2}-5w+5\right)$$

Evaluate

$$w\left(8w^{5}+3w^{4}+5\right)$$