Edexcel A Level Further Maths: Core Pure:復(fù)習(xí)筆記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
- Note 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:
- Coefficients of?x:
- Finally, compare the coefficients
- 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
- The sum of the roots
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 you know the values of
-
- If the second quadratic has roots
and 
?, then use the identities:
- If the second quadratic has roots
-
- If the second quadratic has roots
and 
, then use the identities:
- If the second quadratic has roots
- 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 .
?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
- Note 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?x2:
- Coefficients of?x:
- Constant terms:
- Coefficients of?x2:
- You can then equate the two forms:
- 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 roots
- The sum of the product pairs of roots
?is equal to 
- This ‘sum of pairs’ ?can also be denoted by
- This ‘sum of pairs’
- 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!
- The product of roots
- Unless an exam question specifically asks you to prove these results, you can always use them without proof to answer questions about cubics
- The sum of the roots
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
-
- If the second cubic has roots
, then use the identities:
- i.e.,
- If the second cubic has roots
-
- If the second cubic has roots
, 
, and 
, then use the identities:
- If the second cubic has roots
- 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
- This is done by recalling that a cubic with three given roots can be written in the form
Worked Example
a)Given the cubic equation 
, find, , and
, find, , and
b)Another cubic has roots , , and . Find 
.
, , 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:
- (Note that you will not be asked about a quartic with roots
- 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
- Be aware that this will not give a unique answer
Worked Example
a)???The roots of 
?are α、β、γ and δ. ?Find 
, and .
?are α、β、γ and δ. ?Find , and .
b) Another quartic has roots ，， and . Find the value of 
.
，， 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 sum of the roots
- For a quartic with roots
- 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
- In each case the sum or the product of the ‘new roots’ can be linked back to
- 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
- or if the new roots are cubed, then start by considering
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 
.
, 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 
.
?which has roots .
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