Modelling and control by Anthony Rossiter

FACTORISATION AND ROOTS OF POLYNOMIALS

 

This is a section in the chapter on mathematical skills. Some is revision of pre-university mathematics and some is syllabus mostly covered in year 1 of engineering programmes. Use the left hand toolbar to access the other chapters and themes.

It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials, logarithms and Laplace.

This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

  1. Roots and polynomials 1 - roots from a plot.
  2. Roots and polynomials 2 - what is a factor?.
  3. Roots and polynomials 3 - 1st order.
  4. Roots and polynomials 4 - 2nd order.
  5. Roots and polynomials 5 - higher order.
  6. Roots and polynomials 6 - factorising high order polynomials.
  7. Roots and polynomials 7 - poles/zeros and transfer functions.
  8. Roots and polynomials 8 - worked examples and inequalities.
  9. Roots and polynomials 9 - assessing stability and the Routh array .
  10. Roots and polynomials 10 - tutorial sheet and use of MATLAB.

Roots and polynomials 1 - roots from a plot

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction which focuses on understanding what we mean by a root.

Quick test question

A root of y=f(x) corresponds to:
A. A point where the output y equals the input x.
B. Values of y such that x=0.
C. Values of x such that y=0.
D. None of the above.

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Roots and polynomials 2 - what is a factor?

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction which focuses on understanding what we mean by a factor.

Quick test question

Which statements are true for factors of f(x)?
A. The product of all the factors gives f(x).
B. The sum of all the factors gives f(x).
C. Factors and roots are the same thing.
D. None of the above.

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Roots and polynomials 3 - 1st order

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction which focuses on 1st order polynomials.

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Roots and polynomials 4 - 2nd order

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction which focuses on 2nd order polynomials.

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Roots and polynomials 5 - higher order

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction which focuses on high order polynomials.

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Roots and polynomials 6 - factorising high order polynomials

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video  focuses on numerical examples of how to factorise  a polynomial.

Quick test question

When factorising high order polynomials on pen and paper.
A. Long division will be required.
B. An examiner will normally ensure some factors are obvious.
C. The question will always be tedious.
D. None of the above.

Roots and polynomials 7 - poles/zeros and transfer functions

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video is an introduction to concepts of poles/zeros for transfer functions and the meaning of left half plane (LHP) and right half plane (RHP).  

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Quick test question

Left half plane roots are those for which:

A. The imaginary part is less than zero.
B. The real part is positive.
C. The real part is less than zero.
D. None of the above.

Roots and polynomials 8 - worked examples and inequalities

Lectures aimed at understanding what is a root, links to polynomial factors and easy methods for factorising by hand. This video contains a number of worked examples.

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Roots and polynomials 9 - assessing stability and the Routh array

A key reasons for computing poles is due to the strong link with system behaviours and in particular stability. This video looks at simple, back of the envelope calculuations that can be used to determine whether the roots are all in the LHP, or not.

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Roots and polynomials 10 - tutorial sheet and use of MATLAB

Gives a number of tutorial examples for students to attempt before going through the worked solutions. Questions cover factorisation and assessing whether any roots are in the RHP. Also illustrates how MATLAB can be used both to validate your own working and also to determine roots when the computations are difficult by hand.

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