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NSW Y11 Maths Standard NSW Y11 Maths Advanced NSW Y11 Maths Extension 1

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Functions Polynomials Inverse Trig Functions and Identities Rates of Change Permutations and Combinations

NSW Y11 Maths - Extension 1 Polynomials Multiplicity of Roots

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Multiplicity of Roots Theory

NSW Syllabus Reference

NSW Syllabus Reference: ME-F2.2: Sums and products of roots of polynomials. This will require student to 

  • solve problems using the relationships between the roots and coefficients of quadratic, cubic and quartic equations
  • determine the multiplicity of a root of a polynomial equation
  • graph a variety of polynomials and investigate the link between the root of a polynomial equation and the zero on the graph of the related polynomial function

Ref: https://educationstandards.nsw.edu.au/

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Videos

Videos relating to Multiplicity of Roots.

  • Multiplicity of Roots - Video- Multiplicity of Roots

  • Multiplicity of Roots - Video -Multiplicity of Roots 2

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Theory

If \(x = b\) is a root of multiplicity \(r\) of a real polynomial equation \(P(x) = 0\), then \(x = b\) is also a root of the derived polynomial equation \(\dfrac{{dP}}{{dx}} = 0\) of multiplicity \((r - 1)\).

Consider \(P(x) = {(x - b)^r} \times Q(x)\) where \(Q(b) \ne 0\).

\(\dfrac{{dP}}{{dx}} = r{(x - b)^{r - 1}} \times Q(x) + \dfrac{{dQ}}{{dx}} \times {(x - b)^r}\)

\(\dfrac{{dP}}{{dx}} = {(x - b)^{r - 1}}\left\{ {rQ(x) + (x - b)\dfrac{{dQ}}{{dx}}} \right\}\)

\(\therefore \,\,x = b\,\) is a root of multiplicity \((r - 1)\).