Further Pure 1 Roots of Equations

Properties of the roots of cubic equations Cubic equations have roots α, β, γ (gamma) az 3 + bz 2 + cz + d = 0 a(z – α)(z – β)(z – γ) = 0a = 0 This gives the identity az 3 + bz 2 + cz + d = a(z - α)(z - β)(z – γ) Multiplying out az 3 + bz 2 + cz + d = a(z – α)(z – β)(z – γ) = a(z 2 – αz – βz + αβ)(z – γ) = az 3 – a(α + β + γ)z 2 + a(αβ + αγ + βγ)z - aαβγ z2z2 -αz-αz-βz-βzαβ zz3z3 -αz2-αz2 -βz2-βz2 αβz -γ-γ-γz2-γz2 γαzβzγβzγ-αβγ

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?

Properties of the roots of quartic equations Quartic equations have roots α, β, γ, δ (delta) az 4 + bz 3 + cz 2 + dz + e = 0 a(z – α)(z – β)(z – γ)(z – δ) = 0a = 0 This gives the identity az 4 + bz 3 + cz 2 + dz + e = a(z - α)(z - β)(z – γ)(z – δ) Multiplying out (try this yourself) az 4 + bz 3 + cz 2 + dz + e = a(z – α)(z – β)(z – γ)(z – δ) = a(z 2 – αz – βz + αβ)(z 2 – γz – δz + γδ) z2z2 -αz-αz-βz-βzαβ z2z2 z4z4 -αz3-αz3 -βz3-βz3 αβz 2 -γz-γz-γz3-γz3 αγz 2 βγz 2 -αβγz -δz-δz-δz3-δz3 αδz 2 βδz 2 -αβδz γδγδz 2 -αγδz-βγδzαβγδ

Properties of the roots of quartic equations = z 4 – αz 3 – βz 3 – γz 3 – δz 3 + αβz 2 + αγz 2 + βγz 2 + αδz 2 + βδz 2 + γδz 2 – αβγz – αβδz – αγδz – βγδz + αβγδ = z 4 – (α + β + γ + δ)z 3 + (αβ + αγ + βγ + αδ + βδ + γδ)z 2 – (αβγ + αβδ + αγδ + βγδ)z + αβγδ z2z2 -αz-αz-βz-βzαβ z2z2 z4z4 -αz3-αz3 -βz3-βz3 αβz 2 -γz-γz-γz3-γz3 αγz 2 βγz 2 -αβγz -δz-δz-δz3-δz3 αδz 2 βδz 2 -αβδz γδγδz 2 -αγδz-βγδzαβγδ

Properties of the roots of quartic equations Remember the a = a[z 4 – (α + β + γ + δ)z 3 + (αβ + αγ + βγ + αδ + βδ + γδ)z 2 – (αβγ + αβδ + αγδ + βγδ)z + αβγδ] = az 4 – a(α + β + γ + δ)z 3 + a(αβ + αγ + βγ + αδ + βδ + γδ)z 2 – a(αβγ + αβδ + αγδ + βγδ)z + aαβγδ Equating coefficients -a(α + β + γ + δ) = bα + 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 The roots of the equation 2z 3 – 9z 2 – 27z + 54 = 0 form a geometric progression. Find the values of the roots. Remember that an geometric series goes a, ar, ar 2, ……….., ar (n-1) So from this we get α = a, β = ar, γ = ar 2 α + β + γ = -b/a a + ar + ar 2 = 9/2(1) αβ + αγ + βγ = c/aa 2 r + a 2 r 2 + a 2 r 3 =-27/2 (2) αβγ = -d/aa 3 r 3 = -27(3) We can now solve these simultaneous equations.

Example 1 Starting with the product of the roots equation (3). a 3 r 3 = -27 (ar) 3 = -27 ar = -3 Now plug this into equation (1) a + ar + ar 2 = 9/2 (-3/r) (-3/r)r 2 = 9/2 (-3/r) + -15/2 + -3r = 0(-9/2) r – 6r 2 = 0(×2r) 2r 2 + 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 2z 3 – 9z 2 – 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/aa 3 = -27(3) We can now solve these simultaneous equations.

Example 1 – Alternative Algebra Starting with the product of the roots equation (3). a 3 = -27 a = -3 Now plug this into equation (1) a/r + a + ar = 9/2 -3/r r = 9/2 (-3/r) + -15/2 + -3r = 0(-9/2) r – 6r 2 = 0(×2r) 2r 2 + 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 4z 4 + pz 3 + qz 2 - 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 -α 2 + α 2 + αλ + α 2 – αλ – α 2 – αλ – α 2 + αλ + α 2 – λ 2 = q/4 – λ 2 = q/4 q = -4λ 2

Properties of the roots of quintic equations This is only extension but what would be the properties of the roots of a quintic equation? az 5 + bz 4 + cz 3 + dz 2 + 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