<s>[INST] Determine polynomial whose roots are a linear combination of roots of another polynomial Let $\alpha_1, \alpha_2, \alpha_3$ be the roots of the polynomial $p(x)=x^3+5x^2+7x+11$. Find a polynomial whose roots are $\frac{\alpha_1+\alpha_2}{2}, \frac{\alpha_2+\alpha_3}{2}, \frac{\alpha_1+\alpha_3}{2}.$ Calculating the exact roots doesn't really help because only one root is real. I've tried expanding
$$
\left(x-\frac{\alpha_1+\alpha_2}{2}\right)\left(x-\frac{\alpha_2+\alpha_3}{2}\right)\left(x-\frac{\alpha_1+\alpha_3}{2}\right)
$$
but it gets kind of messy. Any "nicer" ideas?
 [/INST] $\bf{Alternatively::}$ If $x=\alpha\;,x=\beta\;,x=\gamma$ are the roots
of the equation $x^3+5x^2+7x+11=0.\;,$ Then $\alpha+\beta+\gamma = -5$
Now have to find an equation whose roots are $\displaystyle \frac{\alpha+\beta}{2}$ and $\displaystyle \frac{\beta+\gamma}{2}$ and $\displaystyle \frac{\gamma+\alpha}{2}$
Now Using above relation $$\alpha+\beta+\gamma = -5$$
We can write $$\alpha+\beta = -5-\gamma$$ and $$\beta+\gamma = -5-\alpha$$ and $\gamma+\alpha = -5-\beta$
So now we have to find an equation whose roots are $$\displaystyle \frac{-5-\alpha}{2}\;\;,\frac{-5-\beta}{2}\;\;,\frac{-5-\gamma}{2}.$$
so Let $$\displaystyle y = \frac{-5-x}{2}\Rightarrow 2y=-5-x\Rightarrow x=-\left(5+2y\right).$$
Now Put value of $y$ into $x^3+5x^2+7x+11=0\;,$ 
We get $$\displaystyle -(2y+5)^3+5(2y+5)^2-7(2y+5)+11=0$$
So our polynomial equation is $$(2y+5)^3-5(2y+5)^2+7(2y+5)-11=0$$
So we get $$8y^3+40y^2+64y+24 = 0\Rightarrow y^3+5y^2+8y+3=0$$
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