Solving polynomial equations by iteration

Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

?
Intros
Start Watching
Lessons
  1. Introduction to solving polynomial equations by iteration
  2. Direct/Fixed point iteration
  3. Iteration by bisection
  4. Newton-Raphson method
?
Examples
Start Watching
Lessons
  1. Solving Equations Using Direct Iteration
    1. Show that x2−5x−8=0x^2-5x-8=0 can be written in the form x=8+5xx=\sqrt{8+5x}.
    2. Use the iteration formula xn+1=8+5xnx_{n+1}=\sqrt{8+5x_n} to find x3x_3 to 22 decimal places. Start with x0=2x_0=2.
  2. Solving Equations Using Direct Iteration
    1. Show that x3−x−8=0x^3-x-8=0 can be written in the form x=3x+8x={^3}\sqrt{x+8}.
    2. Use the iteration formula xn+1=3xn+8x_{n+1}={^3}\sqrt{x_n+8} to find x4x_4 to 22 decimal places. Start with x1=0x_1=0.
  3. Evaluating equations Using Iteration by Bisection
    The equation x3+5x−7=91x^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 x3−x2=39x^3-x^2=39. Give your answer to 1 decimal place.
      1. Analyzing Equations Using Newton-Raphson Method
        Given x2−6x+5=0x^2-6x+5=0.
        1. Find the iteration formula.
        2. Use the iteration formula found in (a) to approximate the solution. Start with x1=2x_1=2.
      Free to Join!
      StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.

      • Easily See Your Progress

        We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.

      • Make Use of Our Learning Aids

        Last Viewed
        Practice Accuracy
        Suggested Tasks

        Get quick access to the topic you're currently learning.

        See how well your practice sessions are going over time.

        Stay on track with our daily recommendations.

      • Earn Achievements as You Learn

        Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.

      • Create and Customize Your Avatar

        Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
      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 xn+1x_{n+1}.
      4. The RHS becomes xnx_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:
      xn+1=xn−f(xn)f′(xn)x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}
      Basic Concepts
      ?