Solve equations

task1 You are given n, find roots of the polynomial \(x^n + x^{n - 1} + \cdots + x + 1\), that is a polynomial of degree \(n\), that has all coefficints equal to 1. Return roots as one row-matrix.
- Make the function task1 also plot the roots on a complex plane. Just call plot for the matrix with roots.
- comment out the plot that you have just made. Extract real and imaginary parts of the roots and then plot them again by supplying real-valued x and y coordinates of the roots.
task2 Solve the equation \(x + \sin(x)=1\).
task3 solve the system of equations \(\begin{cases} x + \sin(y) = 1 \\ y + \cos(x) = 1 \end{cases}\)
task4 You are given a real number a. Solve the equation \(e^x - e^{-x}=a\). Return for results from the function:
- Solve with fsolve.
- Solve with fzero, you will need to specify a segment where to search for the solution.
- Solve with an explicit expression. Think, how to express x in terms of a.
- The difference between solutions with the explicit expression and with fsolve.
test your function on a = 1 000 000.
task_circles The circle is specified with a matrix of three values: [x, y, r], that is its center and its radius. You are given two circles. Find their intersections. To do this, create a system of two equations, and use fsolve to solve it. Your function should return a matrix with two rows, where intersection points are defined by columns. The function should also draw both circles and use a special marker to mark intersections. Don’t use an explicit expression for evaluating intersections and don’t do any geometry. Just use fsolve.
- To start, find at least one intersection point
- Then find both points. Think, what initial points should be specified for fsolve.