Numerical root finding for 5th degree polyomial

Question

I have the equation $y^5 -ay -b=0$.

I need to get a solution whether numerical or analytical. I heard $5$th order polynomials are not solvable analytically, so how can I get the root numerically. Please guide me, I am new to numerical analysis.

Answer 1

Numerical method

The best suited numerical method is newtons method.

Here's the iteration

$$x_{n+1}=x_n-\dfrac{x_n^{~5}-ax_n-b}{5x_n^{~4}-a}\newcommand{\BR}{\operatorname{BR}}$$

Simply choose a guess for the solution of the equation and call it $x_0$ then using the above formula, calculate as many iterations you want, each iteration get's you closer to the actual value. (it's worth noting that if you made a bad guess, the formula might not converge)

Analytic method

We can express the solution of your equation analytically with the function $\BR(x)$ as the formula below^[1] $$\sqrt[4]{\dfrac{a}{5}}\cdot\BR\left(\frac b4\cdot\sqrt[4]{\frac{25}{a^5}}\right)$$

Where $\BR(x)$ is the inverse of $x^5+x$, this inverse has a taylor series of^[2]

$$\BR(x)=-\sum_{k=0}^\infty\binom{5k}{k}\dfrac{(-1)^ka^{4k+1}}{4k+1}$$