Further Pure 1

Lesson 10 – Roots of Equations

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Properties of the roots of cubic equations 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)

= az3 – a(α + β + γ)z2 + a(αβ + αγ + βγ)z - aαβγ

z2 -αz -βz αβ

z z3 -αz2 -βz2 αβz

-γ -γz2 γαz βzγ -αβγ

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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 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 -αz -βz αβ

z2 z4 -αz3 -βz3 αβz2

-γz -γz3 αγz2 βγz2 -αβγz

-δz -δz3 αδz2 βδz2 -αβδz

γδ γδz2 -αγδz -βγδz αβγδ

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Properties of the roots of quartic equations

= 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 z4 -αz3 -βz3 αβz2

-γz -γz3 αγz2 βγz2 -αβγz

-δz -δz3 αδz2 βδz2 -αβδz

γδ γδz2 -αγδz -βγδz αβγδ

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Properties of the roots of quartic equations

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

WiltshireExample 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 α = a, β = ar, γ = ar2

α + β + γ = -b/a a + ar + ar2 = 9/2 (1)

αβ + αγ + βγ = c/a a2r + a2r2 + a2r3 =-27/2 (2)

αβγ = -d/a a3r3 = -27 (3) We can now solve these simultaneous equations.

WiltshireExample 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

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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 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.

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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

WiltshireExample 2

The roots of the quartic equation 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

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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