Math Tutorial 11.3 - Solutions for Polynomial Equations

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There are 5 lessons in this math tutorial covering Solutions for Polynomial Equations. The tutorial starts with an introduction to Solutions for Polynomial Equations and is then followed with a list of the separate lessons, the tutorial is designed to be read in order but you can skip to a specific lesson or return to recover a specific math lesson as required to build your math knowledge of Solutions for Polynomial Equations. you can access all the lessons from this tutorial below.

In this Math tutorial, you will learn:

  • What is the synthetic division method of polynomials?
  • What are the components of the division of two polynomials?
  • What is the procedure used in the division of polynomials when using the Synthetic Division Method?
  • What are the roots of a polynomial? How to find them?
  • What does the Rational Root (Zero) Theorem say?
  • Why the Rational Root (Zero) Theorem is useful in solving polynomial equations?
  • How to list all possible roots (zeroes) of a polynomial equation?
  • How to identify the zeroes of a polynomial from the whole list of possible zeroes?
  • How to solve polynomial equations?


So far, we have reviewed several situations where the problem asked us to calculate the zeroes of a polynomial. We completed these in a number of ways. In first-degree polynomials, this was an easy task as these polynomials have a single zero (the root of the corresponding linear equation with one variable). In second-degree polynomial, we used either factorisation or the quadratic formula to find the two possible zeroes. In third-degree polynomials, we mostly suggested the iterative methods to find the roots, as they rarely correspond to whole numbers. In other cases, we divided the given polynomial by another supposedly polynomial factor by applying the polynomial division rules.

Obviously, all of these methods have their limitations and are used only in specific contexts. It is important therefore to find a method that is more comprehensive and usable. This method is called "the Synthetic Division Method of Polynomials" - a method that we are going to present in this tutorial. This method is combined with an important algebra theorem known as the "Rational Root (or Rational Zero) Theorem", which is used to prove the correctness of the division method mentioned above. When combined, the above methods help to identify the zeroes of any polynomial with integer coefficients and to find all possible factorisation of the original polynomial.

Identifying the zeroes of a polynomial P(x) means finding the solution of the polynomial equation P(x) = 0, which is the main objective of this tutorial.

Please select a specific "Solutions for Polynomial Equations" lesson from the table below, review the video tutorial, print the revision notes or use the practice question to improve your knowledge of this math topic.

Polynomials Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
11.3Solutions for Polynomial Equations
Lesson IDMath Lesson TitleLessonVideo
11.3.1The Synthetic Division Method of Polynomials
11.3.2The Rational Zero (Root) Theorem
11.3.3How to list all Possible Zeroes (Roots) of a Polynomial Equation
11.3.4How to identify the Zeroes of the Polynomial from the List of Possible Zeroes
11.3.5How to Solve Polynomial Equations

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Enjoy the "Solutions for Polynomial Equations" math tutorial? People who liked the "Solutions for Polynomial Equations" tutorial found the following resources useful:

  1. Math tutorial Feedback. Helps other - Leave a rating for this tutorial (see below)
  2. Polynomials Video tutorial: Solutions for Polynomial Equations. Watch or listen to the Solutions for Polynomial Equations video tutorial, a useful way to help you revise when travelling to and from school/college
  3. Polynomials Revision Notes: Solutions for Polynomial Equations. Print the notes so you can revise the key points covered in the math tutorial for Solutions for Polynomial Equations
  4. Polynomials Practice Questions: Solutions for Polynomial Equations. Test and improve your knowledge of Solutions for Polynomial Equations with example questins and answers
  5. Check your calculations for Polynomials questions with our excellent Polynomials calculators which contain full equations and calculations clearly displayed line by line. See the Polynomials Calculators by iCalculator™ below.
  6. Continuing learning polynomials - read our next math tutorial: Rational Expressions

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