Are there examples of polynomials $x\_1(t), x\_2(t) \in \mathbb{Q}[t]$ of equal degree at least one, with $\gcd(x\_1(t), x\_2(t)) = 1$, such that the sum $(x\_1(t))^4 + (x\_2(t))^4$ is divisible by the square of a polynomial $z(t)$ which is defined over $\mathbb{Q}$? If this question is too much to ask for, can we find examples such that $(x\_1(t))^4 + (x\_2(t))^4$ has a double root in $\mathbb{C}$? 


Edit: in view of the answer given by Jarek Kuben, I am refining the question to the following: suppose we want the degree of $x\_1, x\_2$ to be at most $3$. Can one produce examples still?



Examples of polynomials $x_1(t), x_2(t)$ such that $(x_1(t))^4 + (x_2(t))^4$ has a double root
Here's an example: $x\_1(t)=3t^4 + 4t - 1$, $x\_2(t)=t^4 - 4t^3 - 3$. Then


$\begin{align\*}(x\_1&(t))^4 + (x\_2(t))^4=\\
&82t^{16} - 16t^{15} + 96t^{14} + 176t^{13} + 136t^{12} + 144t^{11} + 288t^{10} + 336t^9 +\\
&108t^8 + 336t^7 + 288t^6 + 144t^5 + 136t^4 + 176t^3 +96t^2 - 16t + 82=\\
&2\cdot(t^4 + 1)^2 \cdot (41t^8 - 8t^7 + 48t^6 + 88t^5 - 14t^4 + 88t^3 + 48t^2 - 8t + 41).
\end{align\*}$


Also $x\_1(t)$ and $x\_2(t)$ are coprime since $\operatorname{Res}(x\_1(t),x\_2(t))=-3072\ne0$.


EDIT: Such examples can be found by factorization described by Joe Silverman. Simply choose some $f(t)\in\mathbb{Q}(\zeta)[t]$, where $\zeta=e^{2\pi i/8}$ (in the example above it's $f(t)=t-\zeta$) and look for non-zero $g(t)\in\mathbb{Q}(\zeta)[t]$ such that $f(t)^2g(t)$ is of the form $x\_1(t)-\zeta x\_2(t)$ for some $x\_1(t),x\_2(t)\in\mathbb{Q}[t]$ (in this example it turns out that $g(t)$ must be at least quadratic).


As for the refined question, for quadratic $x\_1(t)$ and $x\_2(t)$ it's easy to calculate that $x\_1(t)-\zeta x\_2(t)$ has multiple root only if both $x\_1(t)$ and $x\_2(t)$ have the same multiple root, thus the answer is no. For cubic polynomials it seems much more difficult, it leads to the question whether certain hypersurface in $\mathbb{P}^2$ of degree $8$ has any rational points.


