$$\frac{150}{100} = 1+x+x^2+x^3+x^4+x^5$$
Answer
$$ \begin{matrix}x_1 = 0.33426 & x_2 = -0.9353+0.64221i & x_3 = -0.9353-0.64221i \\[1 em] x_4 = 0.26817+1.0441i & x_5 = 0.26817-1.0441i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{150}{100} &= 1+x+x^2+x^3+x^4+x^5&& \text{multiply ALL terms by } \color{blue}{ 100 }. \\[1 em]100\cdot\frac{150}{100} &= 100\cdot1+100x+100x^2+100x^3+100x^4+100x^5&& \text{cancel out the denominators} \\[1 em]150 &= 100+100x+100x^2+100x^3+100x^4+100x^5&& \text{simplify right side} \\[1 em]150 &= 100x^5+100x^4+100x^3+100x^2+100x+100&& \text{move all terms to the left hand side } \\[1 em]150-100x^5-100x^4-100x^3-100x^2-100x-100 &= 0&& \text{simplify left side} \\[1 em]-100x^5-100x^4-100x^3-100x^2-100x+50 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
This page was created using
Polynomial Equations Solver