Question

Find all real solutions of the polynomial equation. $$x^{5}-x^{4}-3 x^{3}+5 x^{2}-2 x=0$$

Short answer

The solution to the equation is \(x=0\) and the roots of the polynomial \(x^{4}-x^{3}-3x^{2}+5x-2=0\). The roots must be calculated separately using a suitable method such as synthetic division or a graphing tool.

Step-by-step solution

Rearrange the equation

Rearrange the equation by factoring out the common term x. The equation can be rewritten as: \(x(x^{4}-x^{3}-3x^{2}+5x-2)=0\)

Solve the factored equations

Because the product of these two factors equals zero, one or both must be equal to zero. Set each factor equal to zero and solve. So we have: 1) \(x=0\)2) \(x^{4}-x^{3}-3x^{2}+5x-2=0\). The second equation is a fourth degree polynomial equation, one might proceed by attempting synthetic division or using a graphing tool or calculator to find the roots, as this is not easily solvable by manual calculational methods.

Find solutions for the factored equations

Our first solution from the first equation is \(x=0\). The solutions of the second equation are the real roots of that polynomial. Assume these roots to be \(a, b, c, d\). They might be real or complex.