Question

If ALPHA, BETA, GAMMA are zeros of polynomial
$x {}^{3} - 3x + 1$
then find the value of
$( \alpha + \beta ) {}^{3} + ( \beta + \gamma ) {}^{3} + ( \gamma + \alpha ) {}^{3}$
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Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{p(x)=x^3-3x+1}$

Its zeros are $\alpha,\,\beta,\,\gamma$

Now,

$\tt{Sum\,\,of\,\,roots,\,\alpha+\beta+\gamma=0}\\\\\tt{Sum\,\,of\,\,roots\,\,taken\,\,two\,\,at\,\,a\,\,time,\,\alpha\beta+\beta\gamma+\gamma\alpha=-3}\\\\\tt{Product\,\,of\,\,roots,\,\alpha\beta\gamma=-1}$

$\tt{\bigstar\,\,\,We\,\,\,know,\,\,\,a^3+b^3+c^3=3abc,\,\,if\,\,a+b+c=0}$

We are given to find

$\sf{S=\left(\alpha+\beta\right)^3+\left(\beta+\gamma\right)^3+\left(\gamma+\alpha\right)^3}$

$\sf{Now,\,\,consider\,\,that\,\,(\alpha+\beta)+(\beta+\gamma)+(\gamma+\alpha)=2(\alpha+\beta+\gamma)=0}$

So,

$\sf{\left(\alpha+\beta\right)^3+\left(\beta+\gamma\right)^3+\left(\gamma+\alpha\right)^3=3\cdot\left(\alpha+\beta\right)\cdot\left(\beta+\gamma\right)\cdot\left(\gamma+\alpha\right)}$

$\sf{\implies\,S=3\cdot\left(\alpha\beta+\beta^2+\gamma\alpha+\beta\gamma\right)\cdot\left(\gamma+\alpha\right)}$

$\sf{\implies\,S=3\cdot\left(\alpha\beta\gamma+\beta^2\gamma+\gamma^2\alpha+\beta\gamma^2+\alpha^2\beta+\alpha\beta^2+\alpha^2\gamma+\alpha\beta\gamma\right)}$

$\sf{\implies\,S=3\cdot\left(2\alpha\beta\gamma+\beta^2\gamma+\gamma^2\alpha+\beta\gamma^2+\alpha^2\beta+\alpha\beta^2+\alpha^2\gamma\right)}$

$\sf{\implies\,S=3\cdot\left(2\alpha\beta\gamma+\gamma^2(\alpha+\beta)+\alpha^2(\beta+\gamma)+\beta^2(\alpha+\gamma)\right)}$

$\sf{\implies\,S=3\cdot\left(2\alpha\beta\gamma+\gamma^2(-\gamma)+\alpha^2(-\alpha)+\beta^2(-\beta)\right)}$

$\sf{\implies\,S=3\cdot\left(2\alpha\beta\gamma-\gamma^3-\alpha^3-\beta^3\right)}$

$\sf{\implies\,S=3\cdot\left(2\alpha\beta\gamma-\alpha^3-\beta^3-\gamma^3\right)}$

Again, we have $\tt{\alpha+\beta+\gamma=0}$

So,

$\sf{\implies\,S=3\cdot\left(2\alpha\beta\gamma-3\,\alpha\beta\gamma\right)}$

$\sf{\implies\,S=3\cdot\left(-\alpha\beta\gamma\right)}$

$\sf{\implies\,S=-3\alpha\beta\gamma}$

$\sf{\implies\,S=-3(-1)=3}$