Roots of Polynomials (Edexcel A Level Further Maths: Core Pure)

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Jamie W

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Jamie W

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Roots of Quadratics

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

  • Because a quadratic equation a x squared plus b x plus c equals 0  (where a not equal to 0) has roots alpha and beta, you can write this equation instead in the form a left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis equals 0
      • Note that left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis equals x squared minus left parenthesis alpha plus beta right parenthesis x plus alpha beta
      • It is possible that the roots are repeated, i.e. that alpha equals beta
    • You can then equate the two forms:
      • a x squared plus b x plus c equals a left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis
    • Then (because a not equal to 0) you can divide both sides of that by a and expand the brackets:
      • x squared plus b over a x plus c over a equals x squared minus left parenthesis alpha plus beta right parenthesis x plus alpha beta
    • Finally, compare the coefficients
      • Coefficients of xnegative b over a equals left parenthesis alpha plus beta right parenthesis
      • Constant terms: c over a equals space alpha beta
  • Therefore for a quadratic equation a x squared plus b x plus c equals 0  :
    • The sum of the roots alpha plus beta is equal to negative b over a
    • The product of the roots alpha beta is equal to c over a
    • 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 alpha beta and alpha plus beta(where alpha and beta are the roots of the first quadratic)
    • If you know the values of alpha and beta 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 alpha squared and beta squared, then use the identities:
      • left parenthesis alpha plus beta right parenthesis squared equals alpha squared plus beta squared plus 2 alpha beta or alpha squared plus beta squared equals left parenthesis alpha plus beta right parenthesis squared minus 2 alpha beta
      • left parenthesis alpha beta right parenthesis squared equals alpha squared beta squared
    • If the second quadratic has roots alpha cubed and beta cubed , then use the identities:
      • left parenthesis alpha plus beta right parenthesis cubed equals alpha cubed plus beta cubed plus 3 alpha beta left parenthesis alpha plus beta right parenthesis
      • left parenthesis alpha beta right parenthesis cubed equals alpha cubed beta cubed
    • If the second quadratic has roots 1 over alpha and 1 over beta, then use the identities:
      • 1 over alpha plus 1 over beta equals fraction numerator alpha plus beta over denominator alpha beta end fraction
      • 1 over alpha cross times 1 over beta equals fraction numerator 1 over denominator alpha beta end fraction
  • 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 a x squared plus b x plus c equals 0 are alpha equals negative 1 half and beta equals 3.

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

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

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Roots of Cubics

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

  • Because a cubic equation a x cubed plus b x squared plus c x plus d equals 0 (where a not equal to 0) has roots alpha, beta and gamma, you can write this equation instead in the form a left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis left parenthesis x minus gamma right parenthesis equals 0
      • Note that left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis left parenthesis x minus gamma right parenthesis equals x cubed minus left parenthesis alpha plus beta plus gamma right parenthesis x squared plus left parenthesis alpha beta plus beta gamma plus alpha gamma right parenthesis x minus alpha beta gamma
      • 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:
      • a x cubed plus b x squared plus c x plus d equals a left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis left parenthesis x minus gamma right parenthesis
    • Then (because a not equal to 0) you can divide both sides of that by a and expand the brackets:
      • x cubed plus b over a x squared plus c over a x plus d over a equals x cubed minus left parenthesis alpha plus beta plus gamma right parenthesis x squared plus left parenthesis alpha beta plus beta gamma plus alpha gamma right parenthesis x minus alpha beta gamma
    • Finally, compare the coefficients
      • Coefficients of x2negative b over a equals alpha plus beta plus gamma
      • Coefficients of xc over a equals alpha beta plus beta gamma plus alpha gamma
      • Constant terms: negative d over a equals alpha beta gamma
  • Therefore for a cubic equation a x cubed plus b x squared plus c x plus d equals 0 :
    • The sum of the roots alpha plus beta plus gamma is equal to negative b over a  
      • The sum of roots alpha plus beta plus gamma can also be denoted by sum from blank to blank of alpha
    • The sum of the product pairs of roots alpha beta plus beta gamma plus alpha gamma is equal to c over a
      • This ‘sum of pairs’ alpha beta plus beta gamma plus alpha gamma can also be denoted by sum from blank to blank of alpha beta
    • The product of the roots alpha beta gamma is equal to negative d over a
      • The product of roots alpha beta gamma spacecan also be denoted by sum from blank to blank of alpha beta gamma
      • 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 sum from blank to blank of alpha, sum from blank to blank of alpha beta, and sum from blank to blank of alpha beta gamma  (where alpha, beta and gamma are the roots of the first cubic)
    • If you know the values of alphabeta, and gamma 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 alpha squared, beta squared, and gamma squared, then use the identities:
      • left parenthesis alpha plus beta plus gamma right parenthesis squared equals alpha squared plus beta squared plus gamma squared plus 2 left parenthesis alpha beta plus beta gamma plus alpha gamma right parenthesis
      • i.e., left parenthesis alpha plus beta plus gamma right parenthesis squared equals alpha squared plus beta squared plus gamma squared plus 2 sum alpha beta
      • left parenthesis alpha beta gamma right parenthesis squared equals alpha squared beta squared gamma squared
    • If the second cubic has roots alpha cubed comma beta cubed comma gamma cubed, then use the identities:
      • left parenthesis alpha plus beta plus gamma right parenthesis cubed equals alpha cubed plus beta cubed plus gamma cubed plus 3 left parenthesis alpha plus beta plus gamma right parenthesis left parenthesis alpha beta plus beta gamma plus alpha gamma right parenthesis minus 3 alpha beta gamma
      • i.e., left parenthesis alpha plus beta plus gamma right parenthesis cubed equals alpha cubed plus beta cubed plus gamma cubed plus 3 sum alpha sum alpha beta minus 3 sum alpha beta gamma
      • left parenthesis alpha beta gamma right parenthesis cubed equals alpha cubed beta cubed gamma cubed
    • If the second cubic has roots 1 over alpha, 1 over beta, and 1 over gamma, then use the identities:
      • 1 over alpha plus 1 over beta plus 1 over gamma equals fraction numerator alpha beta plus beta gamma plus gamma alpha over denominator alpha beta gamma end fraction
      • 1 over alpha cross times 1 over beta cross times 1 over gamma equals fraction numerator 1 over denominator alpha beta gamma end fraction
  • 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 x cubed minus left parenthesis text sum of roots end text right parenthesis x squared plus left parenthesis text sum of product pairs end text right parenthesis x minus left parenthesis text product of roots end text right parenthesis equals 0
    • 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 x cubed plus 3 x squared minus 10 x minus 24 equals 0, find sum from blank to blank of alphasum from blank to blank of alpha beta, and sum from blank to blank of alpha beta gamma

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b)   Another cubic has roots 1 over alpha, 1 over beta, and 1 over gamma. Find 1 over alpha plus 1 over beta plus 1 over gamma.

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Roots of Quartics

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

  • Because a quartic equation a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0 (where a not equal to 0) has roots alpha, beta, gamma and delta, you can write this equation instead in the form a left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis left parenthesis x minus gamma right parenthesis left parenthesis x minus delta right parenthesis equals 0
      • Note that left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis left parenthesis x minus gamma right parenthesis left parenthesis x minus delta right parenthesis expands to x to the power of 4 minus left parenthesis alpha plus beta plus gamma plus delta right parenthesis x cubed plus left parenthesis alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta right parenthesis x squared minus left parenthesis alpha beta gamma plus alpha beta delta plus alpha gamma delta plus beta gamma delta right parenthesis x plus alpha beta gamma delta
      • 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:
      • a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals a left parenthesis x minus alpha right parenthesis left parenthesis x minus beta right parenthesis left parenthesis x minus gamma right parenthesis left parenthesis x minus delta right parenthesis
    • Then (because a not equal to 0) you can divide both sides of that by a and expand the brackets:
      • x cubed plus b over a x squared plus c over a x plus d over a equals x to the power of 4 minus left parenthesis alpha plus beta plus gamma plus delta right parenthesis x cubed plus left parenthesis alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta right parenthesis x squared minus left parenthesis alpha beta gamma plus alpha beta delta plus alpha gamma delta plus beta gamma delta right parenthesis x plus alpha beta gamma delta
    • Finally, compare the coefficients
      • Coefficients of x3negative b over a equals alpha plus beta plus gamma plus delta
      • Coefficients of x2c over a equals alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta
      • Coefficients of xnegative d over a equals space alpha beta gamma plus alpha beta delta plus alpha gamma delta plus space beta gamma delta
      • Constant terms: e over a equals space alpha beta gamma delta
  • Therefore for a quartic equation a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0 :
    • The sum of the roots alpha plus beta plus gamma plus delta is equal to negative b over a  
      • The sum of roots alpha plus beta plus gamma plus delta can also be denoted by sum alpha
    • The sum of the product pairs of roots alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta space is equal to c over a
      • This ‘sum of pairs’ alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta space can also be denoted by sum alpha beta
    • The sum of the product triples of roots alpha beta gamma plus alpha beta delta plus alpha gamma delta plus space beta gamma delta is equal to negative d over a
      • This ‘sum of triples’ can also be denoted by sum alpha beta gamma
    • The product of the roots alpha beta gamma delta is equal to e over a
      • The product of roots (or ‘product of fours’) alpha beta gamma delta can also be denoted by sum alpha beta gamma delta
    • Unless an exam question specifically asks you to prove these results, you can always use them without proof to answer questions about quartics

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 sum alphasum alpha betasum alpha beta gamma, and sum alpha beta gamma delta (where alpha, beta,gamma and delta are the roots of the first quartic)
    • If you know the values of alphabetagamma, and delta 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 alpha squaredbeta squaredgamma squared and delta squared, then use the identities:
      • left parenthesis alpha plus beta plus gamma plus delta right parenthesis squared equals alpha squared plus beta squared plus gamma squared plus delta squared plus 2 left parenthesis alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta right parenthesis
      • i.e., left parenthesis alpha plus beta plus gamma plus delta right parenthesis squared equals alpha squared plus beta squared plus gamma squared plus delta squared plus 2 sum alpha beta
      • left parenthesis alpha beta gamma delta right parenthesis squared equals alpha squared beta squared gamma squared delta squared
    • (Note that you will not be asked about a quartic with roots alpha cubed comma beta cubed comma gamma cubed and delta cubed)
    • If the second quartic has roots 1 over alpha1 over beta1 over gamma and 1 over delta, then use the identities:
      • 1 over alpha plus 1 over beta plus 1 over gamma plus 1 over delta equals fraction numerator alpha beta gamma plus alpha beta delta plus alpha gamma delta plus beta gamma delta over denominator alpha beta gamma delta end fraction
      • 1 over alpha cross times 1 over beta cross times 1 over gamma cross times 1 over delta equals fraction numerator 1 over denominator alpha beta gamma delta end fraction
  • 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 x to the power of 4 minus left parenthesis text sum of roots end text right parenthesis x cubed plus left parenthesis text sum of product pairs end text right parenthesis x squared minus left parenthesis text sum of product triples end text right parenthesis x plus left parenthesis text product of roots end text right parenthesis equals 0
    • 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 x to the power of 4 minus 12 x cubed plus 33 x squared plus 38 x minus 168 equals 0 are alpha comma space beta comma space gamma and delta.  Find stack sum alpha comma space with blank below and blank on top stack sum alpha beta comma space with blank below and blank on top stack sum alpha beta gamma with blank below and blank on top, and stack sum alpha beta gamma delta with blank below and blank on top.  

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b)

Another quartic has roots alpha squared comma space beta squared comma space gamma squared and delta squared. Find the value of alpha squared plus beta squared plus gamma squared plus delta squared.

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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 alpha comma beta comma gamma comma delta, for example:
      • The sum of the roots alpha plus beta plus gamma plus delta is denoted by sum alpha
      • The sum of the pairs of roots alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta spaceis denoted by sum alpha beta
      • The sum of the triples of roots alpha beta gamma plus alpha beta delta plus alpha gamma delta plus space beta gamma delta is denoted by sum alpha beta gamma
      • The sum of the sets of fours (in this case just one term) alpha beta gamma delta is denoted by sum alpha beta gamma delta
  • The table below summarises the relationships between the coefficients and roots of quadratics, cubics, and quartics:
 

a x squared plus b x plus c equals 0

Roots alpha comma space beta

a x cubed plus b x squared plus c x plus d equals 0

Roots alpha comma space beta comma space gamma

a x to the power of 4 plus b x cubed plus c x squared plus d x plus e equals 0

Roots alpha comma space beta comma space gamma comma space delta

sum alpha

alpha plus beta

equals space minus b over a

alpha plus beta plus gamma

equals space minus b over a

alpha plus beta plus gamma plus delta

equals space minus b over a

sum alpha beta

alpha beta

equals c over a

alpha beta plus beta gamma plus alpha gamma

equals c over a

alpha beta plus beta gamma plus alpha gamma plus gamma delta plus alpha delta plus beta delta

equals c over a

sum alpha beta gamma  

alpha beta gamma

equals negative d over a

alpha beta gamma plus alpha beta delta plus alpha gamma delta plus space beta gamma delta

equals negative d over a

sum alpha beta gamma delta    

alpha beta gamma delta

equals e over a

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 sum alpha comma space sum alpha beta comma space sum alpha beta gamma, and/or sum alpha beta gamma delta (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 alpha and beta of the original quadratic equation
    • In each case the sum or the product of the ‘new roots’ can be linked back to alpha beta or alpha plus beta for the original equation
Roots of new Equation Useful Identities to find sums and products of new roots
1 over alpha comma space 1 over beta

1 over alpha plus 1 over beta equals fraction numerator alpha plus beta over denominator alpha beta end fraction

1 over alpha cross times 1 over beta equals fraction numerator 1 over denominator alpha beta end fraction

alpha squared comma space beta squared

alpha squared plus beta squared equals left parenthesis alpha plus beta right parenthesis squared minus 2 alpha beta

alpha squared beta squared equals left parenthesis alpha beta right parenthesis squared

alpha cubed comma space beta cubed

alpha cubed plus beta cubed equals left parenthesis alpha plus beta right parenthesis cubed minus 3 alpha beta left parenthesis alpha plus beta right parenthesis

alpha cubed beta cubed equals left parenthesis alpha beta right parenthesis cubed

  • 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 left parenthesis sum alpha right parenthesis squared
    • or if the new roots are cubed, then start by considering left parenthesis sum alpha right parenthesis cubed
    • or if the new roots are reciprocals (i.e., 1 over alpha comma space 1 over beta, etc.), then start by adding the new roots together to form a single algebraic fraction

Examiner Tip

  • Although you may be asked to tackle questions on this topic showing full working and without relying on a calculator, you can still use your calculator to check your work by finding the roots of a polynomial in the polynomial solver
  • You can then use these to check your answers for sums of roots, products of roots, etc.

Worked example

a)
Given a polynomial equation of order 5 (a quintic); a x to the power of 5 plus b x to the power of 4 plus c x cubed plus d x squared plus e x plus f equals 0, make 5 conjectures linking the coefficients a, b, c, d, e, f to its roots alpha comma space beta comma space gamma comma space delta comma space epsilon.

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b)
Test your conjectures on the example: x to the power of 5 minus 15 x to the power of 4 plus 85 x cubed minus 225 x squared plus 274 x minus 120 equals 0 which has roots x equals 1 comma space 2 comma space 3 comma space 4 comma space 5.

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.