# Solving polynomial equations by iteration

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##### 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 $x_1=2$.
###### Topic Notes

In this lesson, we will learn:

• Solving Equations Using Direct Iteration
• Evaluating equations Using Iteration by Bisection
• Analyzing Equations Using Newton-Raphson Method
• Iteration means to repeatedly solving an equation to obtain a result using the result from the previous calculation.
• Direct iteration:
1. Rearrange the original equation such that the term in which the variable with the highest exponent is isolated.
2. Leave the variable on its own on the LHS by performing inverse operation.
3. The LHS becomes $x_{n+1}$.
4. The RHS becomes $x_n$.
• Iteration by bisection:
1. Shrink the interval where the roots lies within 2 equal parts.
2. Decide in which part the solution resides.
3. Repeat the steps until a consistent answer is achieved.
• Newton-Raphson method:
$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$