Name: Transformation of Roots of a Quadratic Equation
Uploaded: 2019-07-19
Duration: 11 min 45 s
Description: Here an equation and its roots are given, and we have to find new quadratic equations with new roots, which are simple multiples or factors of the original r...

Question

Find the sum of the squares and the cubes of the roots of the equation

x^{4} + q x^{2} + r x + s = 0

.

Accepted Answer

Given equation is $x^{4} + q x^{2} + r x + s = 0$ .

Let the roots be $α, β, γ, δ$

then $α + β + γ + δ = 0, αβ + α γ + α δ + β γ + β δ + γ δ = q, αβγ + αβδ + αγδ + βγδ = - r, αβγδ = s$ .

We have to find sum of the squares and the cubes of the roots that is $α^{2} + β^{2} + γ^{2} + δ^{2}$ and $α^{3} + β^{3} + γ^{3} + δ^{3}$ .

Now $∵ α^{2} + β^{2} + γ^{2} + δ^{2} = {(α + β + γ + δ)}^{2} - 2 (αβ + α γ + α δ + β γ + β δ + γ δ)$

$∴ α^{2} + β^{2} + γ^{2} + δ^{2} = 0 - 2 q = - 2 q$

Again,

${(α + β + γ + δ)}^{3} = (α + β + γ + δ) \cdot {(α + β + γ + δ)}^{2}$

$= α^{3} + β^{3} + γ^{3} + δ^{3} + 3 α^{2} β + 3 α^{2} γ + 3 α^{2} δ + 3 β^{2} α + 3 β^{2} γ + 3 β^{2} δ + 3 γ^{2} α + 3 γ^{2} β + 3 γ^{2} δ + 3 δ^{2} α + 3 δ^{2} β + 3 δ^{2} γ + 6 (αβγ + αβδ + αγδ + βγδ)$

$\Rightarrow (α^{3} + β^{3} + γ^{3} + δ^{3}) = - (3 α^{2} β + 3 α^{2} γ + 3 α^{2} δ + 3 β^{2} α + 3 β^{2} γ + 3 β^{2} δ + 3 γ^{2} α + 3 γ^{2} β + 3 γ^{2} δ + 3 δ^{2} α + 3 δ^{2} β + 3 δ^{2} γ + 6 (αβγ + αβδ + αγδ + βγδ)) . . . (1)$

Now, $α^{2} β + α^{2} γ + α^{2} δ + β^{2} α + β^{2} γ + β^{2} δ + γ^{2} α + γ^{2} β + γ^{2} δ + δ^{2} α + δ^{2} β + δ^{2} γ = (α + β + γ + δ) (αβ + α γ + α δ + β γ + β δ + γ δ) - 3 (αβγ + αβδ + αγδ + βγδ) . . . (2)$

Using $(2) in (1)$ , we have,

$\Rightarrow (α^{3} + β^{3} + γ^{3} + δ^{3}) = - 3 ((α + β + γ + δ) (αβ + α γ + α δ + β γ + β δ + γ δ) - 3 (αβγ + αβδ + αγδ + βγδ)) - 6 (αβγ + αβδ + αγδ + βγδ)$

$\Rightarrow (α^{3} + β^{3} + γ^{3} + δ^{3}) = - 3 (α + β + γ + δ) (αβ + α γ + α δ + β γ + β δ + γ δ) + 3 (αβγ + αβδ + αγδ + βγδ)$

$\Rightarrow (α^{3} + β^{3} + γ^{3} + δ^{3}) = - 3 r$

Find the sum of the squares and the cubes of the roots of the equation $x^{4} + q x^{2} + r x + s = 0$ .

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Find the sum of the squares and the cubes of the roots of the equation x4+qx2+rx+s=0.

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Find the sum of the squares and the cubes of the roots of the equation $x^{4} + q x^{2} + r x + s = 0$ .