Download FP1 roots of equations lesson 10

Further Pure 1
Lesson 10 –
Roots of Equations
Properties of the roots of
cubic equations
Wiltshire
 Cubic equations have roots α, β, γ (gamma)
 az3 + bz2 + cz + d = 0
a(z – α)(z – β)(z – γ) = 0
a=0
 This gives the identity
az3 + bz2 + cz + d = a(z - α)(z - β)(z – γ)
 Multiplying out
az3 + bz2 + cz + d = a(z – α)(z – β)(z – γ)
= a(z2 – αz – βz + αβ)(z – γ)
z2
-αz
-βz
αβ
z
z3
-αz2
-βz2
αβz
-γ
-γz2
γαz
βzγ
-αβγ
= az3 – a(α + β + γ)z2 + a(αβ + αγ + βγ)z - aαβγ
Properties of the roots of
cubic equations
Equating coefficients
 -a(α + β + γ) = b
α + β + γ = -b/a
 a(αβ + αγ + βγ) = c
αβ + αγ + βγ = c/a
 -aαβγ = d
αβγ = -d/a
 Can you notice a pattern?
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Properties of the roots of
quartic equations
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 Quartic equations have roots α, β, γ, δ (delta)
 az4 + bz3 + cz2 + dz + e = 0
a(z – α)(z – β)(z – γ)(z – δ) = 0
a=0
 This gives the identity
az4 + bz3 + cz2 + dz + e = a(z - α)(z - β)(z – γ)(z – δ)
 Multiplying out (try this yourself)
az4 + bz3 + cz2 + dz + e = a(z – α)(z – β)(z – γ)(z – δ)
= a(z2 – αz – βz + αβ)(z2 – γz – δz + γδ)
z2
-γz
-δz
γδ
z2
z4
-γz3
-δz3
γδz2
-αz
-αz3
αγz2
αδz2
-αγδz
-βz
-βz3
βγz2
βδz2
-βγδz
αβ
αβz2
-αβγz
-αβδz
αβγδ
Properties of the roots of
quartic equations
Wiltshire
 = z4 – αz3 – βz3 – γz3 – δz3 + αβz2 + αγz2 + βγz2 + αδz2
+ βδz2 + γδz2 – αβγz – αβδz – αγδz – βγδz + αβγδ
 = z4 – (α + β + γ + δ)z3 + (αβ + αγ + βγ + αδ + βδ + γδ)z2
– (αβγ + αβδ + αγδ + βγδ)z + αβγδ
z2
-αz
-βz
αβ
z2
-γz
z4
-γz3
-αz3
αγz2
-βz3
βγz2
αβz2
-αβγz
-δz
γδ
-δz3
γδz2
αδz2
-αγδz
βδz2
-βγδz
-αβδz
αβγδ
Properties of the roots of
quartic equations
Wiltshire
 Remember the a
 = a[z4 – (α + β + γ + δ)z3 + (αβ + αγ + βγ + αδ + βδ + γδ)z2






– (αβγ + αβδ + αγδ + βγδ)z + αβγδ]
= az4 – a(α + β + γ + δ)z3 + a(αβ + αγ + βγ + αδ + βδ +
γδ)z2 – a(αβγ + αβδ + αγδ + βγδ)z + aαβγδ
Equating coefficients
-a(α + β + γ + δ) = b
α + β + γ + δ = -b/a = Σα
a(αβ + αγ + βγ + αδ + βδ + γδ) = c
αβ + αγ + βγ + αδ + βδ + γδ = c/a = Σαβ
-a(αβγ + αβδ + αγδ + βγδ) = d
αβγ + αβδ + αγδ + βγδ = -d/a = Σαβγ
aαβγδ = e
αβγδ = e/a
Example 1
Wiltshire
 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 α = a, β = ar, γ = ar2
2 = 9/2
α + β + γ = -b/a
a
+
ar
+
ar
(1)

αβ + αγ + βγ = c/a  a2r + a2r2 + a2r3 =-27/2
(2)
3 3
αβγ = -d/a
(3)
 a r = -27
We can now solve these simultaneous equations.
Example 1
Wiltshire
 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
Example 1 – Alternative Algebra
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 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 α = a/r, β = a, γ = ar
α + β + γ = -b/a
a/r + a + ar = 9/2
(1)
 We have ignored equation 2 because it did not help last
time.
αβγ = -d/a
a3 = -27
(3)
 We can now solve these simultaneous equations.
Example 1 – Alternative Algebra
Wiltshire
 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
Example 2
 The roots of the quartic equation




Wiltshire
4z4 + pz3 + qz2 - z + 3 = 0 are α, -α, α + λ, α – λ where α
& λ are real numbers.
i) Express p & q in terms of α & λ.
α + β + γ + δ = -b/a
α + (-α) + (α + λ) + (α – λ) = -p/4
2α = -p/4
p = -8α
αβ + αγ + αδ + βγ + βδ + γδ = c/a
(α)(-α) + α(α + λ) + α(α - λ) + (-α)(α + λ) + (-α)(α - λ) + (α
+ λ)(α – λ) = q/4
-α2 + α2 + αλ + α2 – αλ – α2 – αλ – α2 + αλ + α2 – λ2 = q/4
– λ2
= q/4
q = -4λ2
Properties of the roots of
quintic equations
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 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