Further Pure 1

1
Further Pure 1

• Lesson 10
• Roots of Equations

2
Properties of the roots of cubic equations

• Cubic equations have roots a, ß, ? (gamma)
• az3 bz2 cz d 0
• a(z a)(z ß)(z ?) 0 a 0
• This gives the identity
• az3 bz2 cz d a(z - a)(z - ß)(z ?)
• Multiplying out
• az3 bz2 cz d a(z a)(z ß)(z ?)
• a(z2 az ßz aß)(z ?)
• az3 a(a ß ?)z2 a(aß a? ß?)z -
  aaß?

3
Properties of the roots of cubic equations

• Equating coefficients
• -a(a ß ?) b
• a ß ? -b/a
• a(aß a? ß?) c
• aß a? ß? c/a
• -aaß? d
• aß? -d/a
• Can you notice a pattern?

4
Properties of the roots of quartic equations

• Quartic equations have roots a, ß, ?, d (delta)
• az4 bz3 cz2 dz e 0
• a(z a)(z ß)(z ?)(z d) 0 a 0
• This gives the identity
• az4 bz3 cz2 dz e a(z - a)(z - ß)(z
  ?)(z d)
• Multiplying out (try this yourself)
• az4 bz3 cz2 dz e a(z a)(z ß)(z
  ?)(z d)
• a(z2 az ßz aß)(z2 ?z dz ?d)
  

5
Properties of the roots of quartic equations

• z4 az3 ßz3 ?z3 dz3 aßz2 a?z2 ß?z2
  adz2 ßdz2 ?dz2 aß?z aßdz a?dz
  ß?dz aß?d
• z4 (a ß ? d)z3 (aß a? ß? ad
  ßd ?d)z2 (aß? aßd a?d ß?d)z aß?d

6
Properties of the roots of quartic equations

• Remember the a
• az4 (a ß ? d)z3 (aß a? ß? ad
  ßd ?d)z2 (aß? aßd a?d ß?d)z aß?d
• az4 a(a ß ? d)z3 a(aß a? ß? ad
  ßd ?d)z2 a(aß? aßd a?d ß?d)z aaß?d
• Equating coefficients
• -a(a ß ? d) b a ß ? d -b/a Sa
• a(aß a? ß? ad ßd ?d) c
• aß a? ß? ad ßd ?d c/a
  Saß
• -a(aß? aßd a?d ß?d) d
• aß? aßd a?d ß?d -d/a Saß?
• aaß?d e aß?d
  e/a

7
Example 1

• The roots of the equation 2z3 9z2 27z 54
  0 form a geometric progression.
• Find the values of the roots.
• Remember that an geometric series goes a, ar,
  ar2, .., ar(n-1)
• So from this we get a a, ß ar, ? ar2
• a ß ? -b/a a ar ar2 9/2 (1)
• aß a? ß? c/a a2r a2r2 a2r3 -27/2
  (2)
• aß? -d/a a3r3 -27 (3)
• We can now solve these simultaneous equations.

8
Example 1

• Starting with the product of the roots equation
  (3).
• a3r3 -27 (ar)3 -27 ar -3
• Now plug this into equation (1)
• a ar ar2 9/2
• (-3/r) -3 (-3/r)r2 9/2
• (-3/r) -15/2 -3r 0 (-9/2)
• -6 -15r 6r2 0 (2r)
• 2r2 5r 2 0 (-3)
• (2r 1)(r 2) 0
• r -0.5 -2
• This gives us the arithmetic series 6, -3, 1.5 or
  1.5, -3, 6

9
Example 1 Alternative Algebra

• 2z3 9z2 27z 54 0
• This time because we know that we are going to
  use the product of the roots we could have the
  first 3 terms of the series as a/r, a, ar
• So from this we get a a/r, ß a, ? ar
• a ß ? -b/a a/r a ar 9/2 (1)
• We have ignored equation 2 because it did not
  help last time.
• aß? -d/a a3 -27 (3)
• We can now solve these simultaneous equations.

10
Example 1 Alternative Algebra

• Starting with the product of the roots equation
  (3).
• a3 -27 a -3
• Now plug this into equation (1)
• a/r a ar 9/2
• -3/r -3 -3r 9/2
• (-3/r) -15/2 -3r 0 (-9/2)
• -6 -15r 6r2 0 (2r)
• 2r2 5r 2 0 (-3)
• (2r 1)(r 2) 0
• r -0.5 -2
• This gives us the arithmetic series 6, -3, 1.5 or
  1.5, -3, 6

11
Example 2

• The roots of the quartic equation 4z4
  pz3 qz2 - z 3 0 are a, -a, a ?, a ?
  where a ? are real numbers.
• i) Express p q in terms of a ?.
• a ß ? d -b/a
• a (-a) (a ?) (a ?) -p/4
• 2a -p/4
• p -8a
• aß a? ad ß? ßd ?d c/a
• (a)(-a) a(a ?) a(a - ?) (-a)(a ?)
  (-a)(a - ?) (a ?)(a ?) q/4
• -a2 a2 a? a2 a? a2 a? a2 a? a2
  ?2 q/4
• ?2 q/4
• q -4?2

12
Properties of the roots of quintic equations

• This is only extension but what would be the
  properties of the roots of a quintic equation?
• az5 bz4 cz3 dz2 ez f 0
• The sum of the roots -b/a
• The sum of the product of roots in pairs c/a
• The sum of the product of roots in threes -d/a
• The sum of the product of roots in fours e/a
• The product of the roots -f/a
• Now do Ex 4c pg 110, Ex 4d pg 113