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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 Roots and Coefficients

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Roots and Coefficients 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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  • Roots and Coefficients - Video- Roots and Coefficients

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  • Roots and Coefficients - Video - Roots and coefficients of polynomial equations

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Theory

Quadratic equations

For the quadratic equation \(a{x^2} + bx + c = 0\). 

  • The sum of the roots \(\alpha + \beta = - \dfrac{b}{a}\)
  • The product of the roots \(\alpha \beta = \dfrac{c}{a}\)

Cubic equations

For the cubic equation \(a{x^3} + b{x^2} + cx + d = 0\).

  • The sum of the roots \(\sum {\alpha = \alpha + \beta + \gamma = - \dfrac{b}{a}} \). 
  • The sum of products of each pair of roots \(\sum {\alpha \beta = \alpha \beta + \alpha \gamma + \beta \gamma = \dfrac{c}{a}} \)
  • The product of the roots \(\alpha \beta \gamma = - \dfrac{d}{a}\)

Quadratic Equations

For the quadratic equation \(a{x^4} + b{x^3} + c{x^2} + dx + e = 0\)

  • The sum of the roots \(\sum {\alpha = \alpha + \beta + \gamma + \delta = - \dfrac{b}{a}} \)
  • The sum of products of each pair of roots \(\sum {\alpha \beta = \alpha \beta + \alpha \gamma + \alpha \delta \, + \beta \gamma \, + \beta \delta \, + \gamma \delta = \dfrac{c}{a}} \)
  • The sum of products of each triplet of roots \(\sum {\alpha \beta \gamma = \alpha \beta \gamma + \alpha \beta \delta \, + \alpha \gamma \delta \, + \beta \gamma \delta = - \dfrac{d}{a}} \)
  • The product of the roots \(\alpha \beta \gamma \delta = \dfrac{e}{a}\)