# Solving polynomial equations by iteration

##### Intros
###### Lessons
1. Introduction to solving polynomial equations by iteration
2. Direct/Fixed point iteration
3. Iteration by bisection
4. Newton-Raphson method
##### Examples
###### Lessons
1. Solving Equations Using Direct Iteration
1. Show that $x^2-5x-8=0$ can be written in the form $x=\sqrt{8+5x}$.
2. Use the iteration formula $x_{n+1}=\sqrt{8+5x_n}$ to find $x_3$ to $2$ decimal places. Start with $x_0=2$.
2. Solving Equations Using Direct Iteration
1. Show that $x^3-x-8=0$ can be written in the form $x={^3}\sqrt{x+8}$.
2. Use the iteration formula $x_{n+1}={^3}\sqrt{x_n+8}$ to find $x_4$ to $2$ decimal places. Start with $x_1=0$.
3. Evaluating equations Using Iteration by Bisection
The equation $x^3+5x-7=91$ has a solution between 4 and 5. Use bisection iteration to find the solution and give the answer to 1 decimal place.
1. Use bisection iteration to solve $x^3-x^2=39$. Give your answer to 1 decimal place.
1. Analyzing Equations Using Newton-Raphson Method
Given $x^2-6x+5=0$.
1. Find the iteration formula.
2. Use the iteration formula found in (a) to approximate the solution. Start with $x_1=2$.