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

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

 

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