[Math] How to calculate $\alpha ‘\beta ‘+\alpha ‘\gamma ‘ + \beta ‘\gamma ‘$ when finding roots for a cubic equation

Question

Given the roots of the cubic equation $x^3+4x^2+3x+2=0$ are $\alpha, \beta, \gamma$, determine the cubic equation with roots $\beta\gamma, \gamma\alpha, \alpha\beta$.

How on earth do I work out what the value of $\alpha '\beta '+\alpha '\gamma ' + \beta '\gamma '$?

I worked out that $\alpha '\ + \beta ' + \gamma ' = 3$.

Thanks 🙂

Answer 1

For typing speed, I hope you don't mind if I use latin letters :)

Let $a,b,c$ the roots. You know that $a+b+c=-4$, $ab+bc+ac=3$ and $abc=-2$. Let $a'=bc$, $b'=ac$, $c'=ab$.

Now you want to know:

• $a'+b'+c'=ab+bc+ab=3$
• $a'b'+b'c'+a'c'=abc^2+a^2bc+ab^2c=abc(a+b+c)=8$
• $a'b'c'=a^2b^2c^2=(abc)^2= 4$

Therefore, your polynomial is $x^3-3x^2+8x-4$