1. Find all polynomials of the form $a x^{3}+b x^{2}+c x+d$, which when divided by the binomial $2 x^{2}+1$ give a remainder of $x+2$ and when divided by the binomial $x^{2}+2$ give a remainder of $2 x+1$.
SOLUTION. When dividing (with remainder) a polynomial of the third degree by a polynomial of the second degree, the quotient is equal to a polynomial of the first degree. The coefficient of the first power in it is equal to the ratio of the coefficients of the highest powers of the dividend and divisor, while the absolute term of the quotient is unknown, which we denote as $e$, resp. $g$ for the two given divisions. For the sought polynomial, the following must hold

$$
\begin{aligned}
& a x^{3}+b x^{2}+c x+d=\left(2 x^{2}+1\right)\left(\frac{a}{2} x+e\right)+x+2=a x^{3}+2 e x^{2}+\left(\frac{a}{2}+1\right) x+e+2 \\
& a x^{3}+b x^{2}+c x+d=\left(x^{2}+2\right)(a x+g)+2 x+1=a x^{3}+g x^{2}+(2 a+2) x+2 g+1
\end{aligned}
$$

By comparing the coefficients of the right sides of the previous two equalities, we get

$$
\begin{aligned}
2 e & =g, \\
\frac{a}{2}+1 & =2 a+2, \\
e+2 & =2 g+1 .
\end{aligned}
$$

The solution to this system is $a=-\frac{2}{3}, g=\frac{2}{3}, e=\frac{1}{3}$. Therefore, the problem is satisfied by the unique polynomial

$$
a x^{3}+b x^{2}+c x+d=-\frac{2}{3} x^{3}+\frac{2}{3} x^{2}+\frac{2}{3} x+\frac{7}{3} .
$$

GUIDING PROBLEMS:

N1. Determine the quotient and remainder when dividing the polynomial $2 x^{3}+x^{2}-3 x+5$ by the binomial $x^{2}-x$. Express this fact in the form of an equality without using fractions. $\left[2 x^{3}+x^{2}-3 x+5=\left(x^{2}-x\right) \times\right.$ $\times(2 x+3)+5]$

N2. Determine all quadratic trinomials $a x^{2}+b x+c$ that are divisible without remainder by both the binomial $x-2$ and the binomial $x+1 .\left[a x^{2}-a x-2 a, a \neq 0\right]$

N3. Find all trinomials $p(x)=a x^{2}+b x+c$ that give a remainder of 2 when divided by the binomial $x+1$ and a remainder of 1 when divided by the binomial $x+2$, with $p(1)=61$. [61-C-I-1]

N4. Find all pairs $p, q$ of real numbers such that the polynomial $x^{2}+p x+q$ is a divisor of the polynomial $x^{4}+p x^{2}+q \cdot [56-\mathrm{B}-\mathrm{I}-5]$