B1. The solutions to the equation $8 \cdot 2^{x}=\sqrt[x]{16}$ are the real roots of the polynomial $p(x)=x^{4}-3 x^{3}+a x^{2}+51 x+b$. Calculate $a$ and $b$ and find the other two roots.
B1. Let's solve the exponential equation $2^{x+3}=2^{\frac{4}{x}}$. Equate the exponents $x+3=\frac{4}{x}$. Eliminate the fraction $x^{2}+3 x=4$ and obtain the quadratic equation $x^{2}+3 x-4=0$. The solutions are $x_{1}=-4$ and $x_{2}=1$. Use Horner's algorithm or set up a system of two equations with two unknowns. Solve the system of equations

$$
\begin{array}{r}
-3 a-39=0 \\
a+b+49=0
\end{array}
$$

We get the solutions $a=-13$ and $b=-36$. The quotient when dividing the polynomial $p(x)$ by the polynomial $q(x)=(x+4)(x-1)$, which we obtain from the solutions of the exponential equation, is $x^{2}-6 x+9$. From this, we also get the remaining zeros of the polynomial $x_{3,4}=3$.

Solving the exponential equation

Solution of the quadratic equation $x^{2}+3 x-4=0$

Writing and solving the system of equations

$$
\begin{array}{r}
-3 a-39=0 \\
a+b+49=0
\end{array}
$$