Polynomials (1 Viewer)

senarillosk · Nov 29, 2015

If α, β and γ are the roots of the equation x³ + 5x - 4 = 0, evaluate α³ + β³ + γ³

Need help on this. Thanks!

InteGrand · Nov 29, 2015

senarillosk said:
If α, β and γ are the roots of the equation x³ + 5x - 4 = 0, evaluate α³ + β³ + γ³

Need help on this. Thanks!

$Since $\alpha$ is a root, it satisfies the polynomial equation given, so$

$\begin{align*} \alpha^3 + 5\alpha -4 &=0 \\ \Rightarrow \alpha^3 &= -5 \alpha +4. \quad (1) \end{align*}$

$Sinilarly for the other roots, we have$

$\begin{align*} \beta^3 &= -5\beta + 4 \quad (2) \\ \gamma^3 &= -5\gamma +4. \quad (3) \end{align*}$

$So adding Equations (1), (2) and (3) yields$

$\begin{align*} \alpha^3 +\beta^3 + \gamma^3 &= -5\left( \alpha + \beta + \gamma \right) + 12 \\ &= -5\times 0 +12 \quad (\text{sum of roots of the given polynomial is 0}) \\ &= 12. \end{align*}$

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Polynomials (1 Viewer)

senarillosk

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InteGrand

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