<s>[INST] What condition the roots of the polynomial meet Coefficients of the polynomial $f(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$ meet the condition $a_3^3-4a_2a_3+8a_1=0.$ By substitution $x=y-\frac{a_3}{a_4}$ polynomial $f(x)$ is transformed into a biquadratic polynomial $g(y)=y^4+b_2y^2+b_0,$ where $b_2, b_0$ depend on the polynomial coefficients $f(x).$ What condition the roots of the polynomial $g(y)$ meet. Thanks for your help.
 [/INST] It is given $f(x)=a_4x^4+a_3x^3+a_2x^2+a_1x+a_0$. When we substitute $x=y-\frac{a_3}{a_4}$, we get the following polynomial,
$$f(y)=a_4\left(y-\frac{a_3}{a_4}\right)^4+a_3\left(y-\frac{a_3}{a_4}\right)^3+a_2\left(y-\frac{a_3}{a_4}\right)^2+a_1\left(y-\frac{a_3}{a_4}\right)+a_0$$
Now, we are expanding each term of the polynomial separately.
$$
    \begin{matrix}
    a_4y^4 & - & 4a_3y^3 & + & 6\left(\frac{a_3^2}{a_4}\right)y^2 & - & 4\left(\frac{a_3^3}{a_4^2}\right)y & + & \frac{a_3^4}{a_4^3}\\
     & & a_3y^3 & - & 3\left(\frac{a_3^2}{a_4}\right)y^2 & + & 3\left(\frac{a_3^3}{a_4^2}\right)y & - & \frac{a_3^4}{a_4^3}\\
    & & & & a_2y^2 & - & 2\left(\frac{a_3a_2}{a_4}\right)y & + & \frac{a_3^2a_2}{a_4^2} \\
& & & & & &a_1y & - & \frac{a_3a_1}{a_4}   \\
& & & & & & & & a_0   \\
    \end{matrix}
$$
When we add all these terms together and devide the whole thing by $a_4$, we get,
$$g(y)=y^4-3\frac{a_3}{a_4}y^3+\frac{1}{a_4}\left(3\frac{a_3^2}{a_4}+a_2\right)y^2-\frac{1}{a_4}\left(\frac{a_3^3}{a_4^2}-2\frac{a_3a_2}{a_4}-a_1\right)y+\frac{1}{a_4}\left(\frac{a_3^2a_2}{a_4^2}-\frac{a_3a_1}{a_4}+ a_0\right).$$
Since this transformation is supposed to deliver us the biquadratic polynomial $g(y)=y^4+b_2y^2+b_0,$ we have,
$a_3=0.$
According to the condition satisfied by the coefficients of $f\left(x\right)$, we get,
$a_1=0.$
Therefore, the coefficient of the $2^{nd}$ and $4^{th}$ terms of the polynomial $g\left(y\right)$ vanish giving us the following.
$$g(y)=y^4+\left(\frac{a_2}{a_4}\right)y^2+\frac{a_0}{a_4}$$
If $\beta$ is a root of the equation $g\left(y\right)=0$, then it must meet the following condition.
$$a_4\beta^4+a_2\beta^2+a_0=0.$$
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