Edexcel A Level Further Maths: Core Pure:复习笔记3.1.1 Roots of Polynomials

Roots of Quadratics

How are the roots of a quadratic linked to its coefficients?

  • Because a quadratic equation   (where ) has roots and , you can write this equation instead in the form
      • Note that
      • It is possible that the roots are repeated, i.e. that
    • You can then equate the two forms:
    • Then (because ) you can divide both sides of that by a and expand the brackets:
    • Finally, compare the coefficients
      • Coefficients of x:
      • Constant terms:
  • Therefore for a quadratic equation   :
    • The sum of the roots  is equal to
    • The product of the roots  is equal to
    • Unless an exam question specifically asks you to prove these results, you can always use them without proof to answer questions about quadratics

Related Roots

  • You may be asked to consider two quadratic equations, with the roots of the second quadratic linked to the roots of the first quadratic in some way
    • You are usually required to find the sum or product of the roots of the second equation
  • The strategy is to use identities which contain and(where and are the roots of the first quadratic)
    • If you know the values of and from the first quadratic, you can use them to help find the sum or product of the new roots
    • If the second quadratic has roots and , then use the identities:
    • If the second quadratic has roots and  , then use the identities:
    • If the second quadratic has roots and , then use the identities:
  • You can then form a new equation for a quadratic with the new roots
    • This is done by recalling that a quadratic with a given pair of roots can be written in the form x2 – (sum of the roots)x + (product of the roots) = 0
    • Be aware that this will not give a unique answer
      • This is because multiplying an entire quadratic by a constant does not change its roots
      • You can use this fact, for example, to find a quadratic that has a particular pair of roots AND has all integer coefficients
  • See the worked example below for an example of how to do some of this!

Worked Example

The roots of an equation are and .

a) Find integer values of a, b, and c.

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b) Hence find a quadratic equation whose roots are  and .

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Roots of Cubics

How are the roots of a cubic linked to its coefficients?

  • Because a cubic equation (where ) has roots α, β and , you can write this equation instead in the form
      • Note that
      • It is possible that some of the roots are repeated, i.e. that some or all of them are equal to each other
    • You can then equate the two forms:
    • Then (because ) you can divide both sides of that by a and expand the brackets:
    • Finally, compare the coefficients
      • Coefficients of x2:
      • Coefficients of x:
      • Constant terms:
  • Therefore for a cubic equation  :
    • The sum of the roots  is equal to
      • The sum of roots  can also be denoted by
    • The sum of the product pairs of roots  is equal to
      • This ‘sum of pairs’  can also be denoted by
    • The product of the roots  is equal to
      • The product of roots can also be denoted by
      • See quartic equations where using this ‘sum of triples’ notation makes more sense!
    • Unless an exam question specifically asks you to prove these results, you can always use them without proof to answer questions about cubics

Related Roots

  • You may be asked to consider two cubic equations, with the roots of the second cubic linked to the roots of the first cubic in some way
    • You are usually required to find the sum or product of the roots of the second equation
  • The strategy is to use identities which contain , , and   (where α, β and γ are the roots of the first cubic)
    • If you know the values of α, β, and γ from the first cubic, you can use them to help find the sum or product of the new roots
    • If the second cubic has roots , , and , then use the identities:
      • i.e.,
    • If the second cubic has roots , then use the identities:
      • i.e.,
    • If the second cubic has roots , , and , then use the identities:
  • You can then form a new equation for a cubic with the new roots
    • This is done by recalling that a cubic with three given roots can be written in the form
    • Be aware that this will not give a unique answer
      • This is because multiplying an entire cubic by a constant does not change its roots
      • You can use this fact, for example, to find a cubic that has a particular pair of roots AND has all integer coefficients

Worked Example

a)Given the cubic equation , find, , and

 

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b)Another cubic has roots , , and . Find .

3-1-1-edx-a-fm-we2b-soltn

Roots of Quartics

How are the roots of a quartic linked to its coefficients?

Related Roots

  • You may be asked to consider two quartic equations, with the roots of the second quartic linked to the roots of the first quartic in some way
    • You are usually required to find the sum or product of the roots of the second equation
  • The strategy is to use identities which contain , , , and  (where α, β,γ and δ are the roots of the first quartic)
    • If you know the values of α, β, γ, and δ from the first quartic, you can use them to help find the sum or product of the new roots
    • If the second quartic has roots , , and , then use the identities:
      • i.e.,
    • (Note that you will not be asked about a quartic with roots and )
    • If the second quartic has roots , , and , then use the identities:
  • You can then form a new equation for a quartic with the new roots
    • This is done by recalling that a quartic with four given roots can be written in the form

    • Be aware that this will not give a unique answer
      • This is because multiplying an entire quartic by a constant does not change its roots
      • You can use this fact, for example, to find a quartic that has a particular pair of roots AND has all integer coefficients

Worked Example

a)   The roots of  are α、β、γ and δ.  Find , and .

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b) Another quartic has roots and . Find the value of .

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Roots of Polynomials

What is the general pattern linking the roots to the coefficients of a polynomial?

  • By looking at the links between the coefficients and the roots of quadratics, cubics, and quartics, you can see that a pattern emerges, which also holds true for higher order polynomials
  • It is useful to use sigma notation to keep expressions for sums of roots concise
    • For a quartic with roots , for example:
      • The sum of the roots  is denoted by
      • The sum of the pairs of roots is denoted by
      • The sum of the triples of roots  is denoted by
      • The sum of the sets of fours (in this case just one term)  is denoted by
  • The table below summarises the relationships between the coefficients and roots of quadratics, cubics, and quartics:

How can I find sums and products of related roots?

  • You may be asked to consider a second equation, that has roots linked to the roots of the first equation in some way
    • You are usually required to find the sum or product of the roots of the second equation
  • The strategy is to use identities containing , and/or  (depending on the question and the degree of the polynomial)
    • If you know the value of the roots from the first equation, these identities can help you find the sum or product of the roots of the second equation
  • The table below shows useful identities for finding a new quadratic equation whose roots are related to the roots α and β of the original quadratic equation
    • In each case the sum or the product of the ‘new roots’ can be linked back to or  for the original equation

  • Similar identities that could be useful for cubics and quartics are listed earlier in this revision note in the cubics and quartics sections
  • A good place to start if the new roots are squared, is by considering
    • or if the new roots are cubed, then start by considering
    • or if the new roots are reciprocals (i.e., , etc.), then start by adding the new roots together to form a single algebraic fraction

Worked Example

a) Given a polynomial equation of order 5 (a quintic); , make 5 conjectures linking the coefficients a, b, c, d, e, f to its roots .

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b) Test your conjectures on the example:  which has roots .

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