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

#### 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.

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

### 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

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

### 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 .

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

### 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 .

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