Solve x^5-x^4-5/2x^3+2x^2-1/2x-3

$$x ^ { 5 } - x ^ { 4 } - \frac { 5 } { 2 } x ^ { 3 } + 2 x ^ { 2 } - \frac { 1 } { 2 } x - 3$$

Factor

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

Solution Steps

Factor out \frac{1}{2}.

$$\frac{2x^{5}-2x^{4}-5x^{3}+4x^{2}-x-6}{2}$$

Consider 2x^{5}-2x^{4}-5x^{3}+4x^{2}-x-6. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient 2. One such root is -1. Factor the polynomial by dividing it by x+1.

$$\left(x+1\right)\left(2x^{4}-4x^{3}-x^{2}+5x-6\right)$$

Consider 2x^{4}-4x^{3}-x^{2}+5x-6. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient 2. One such root is 2. Factor the polynomial by dividing it by x-2.

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

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

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

Evaluate

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