Solve 2x^7-x^6-2x^4+x^3+54x-27

$$2 { x }^{ 7 } - { x }^{ 6 } -2 { x }^{ 4 } + { x }^{ 3 } +54x-27$$

Factor

$$\left(2x-1\right)\left(x^{6}-x^{3}+27\right)$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -27 and q divides the leading coefficient 2. One such root is \frac{1}{2}. Factor the polynomial by dividing it by 2x-1. Polynomial x^{6}-x^{3}+27 is not factored since it does not have any rational roots.

$$\left(2x-1\right)\left(x^{6}-x^{3}+27\right)$$

Evaluate

$$\left(2x-1\right)\left(x^{6}-x^{3}+27\right)$$