Let's determine the numbers $a$ and $b$ such that

$$
4 x^{4}+4 x^{3}-11 x^{2}+a x+b
$$

is divisible by $(x-1)^{2}$. The quotient must also be calculated!
I. Solution. If the given fourth-degree expression is divisible by the quadratic $(x-1)^{2}=x^{2}-2 x+1$, the quotient is quadratic, whose first term is $4 x^{2}$, so it is of the form $4 x^{2}+p x+q$. Therefore, it must be:

$$
4 x^{4}+4 x^{3}-11 x^{2}+a x+b \equiv\left(x^{2}-2 x+1\right)\left(4 x^{2}+p x+q\right)
$$

If we perform the multiplication on the right-hand side and arrange the terms according to the powers of $x$, we get:

$$
4 x^{4}+(p-8) x^{3}+(q-2 p+4) x^{2}+(p-2 q) x+q
$$

The identity means that the corresponding coefficients on both sides are equal:

$$
p-8=4, \quad q-2 p+4=-11, \quad p-2 q=a, \quad q=b
$$

From here, step by step: $p=12, q=9, a=-6, b=9$.

The quotient is: $4 x^{2}+12 x+9$.

András Haraszthy (Szent László g. V. o. Bp. X)

II. Solution. The fourth-degree expression is divisible by $(x-1)^{2}$ if it is divisible by $(x-1)$ and the quotient is again divisible by $(x-1)$.

A polynomial expression in $x$ is divisible by $(x-1)$ if substituting $x=1$ results in zero, that is, if

$$
4+4-11+a+b=0, \quad a+b-3 \quad \text { or } \quad b=3-a
$$

Thus, we get: $4 x^{4}+4 x^{3}-11 x^{2}+a x+(3-a)$.

If we divide this by $(x-1)$, the quotient is

$$
4 x^{3}+8 x^{2}-3 x+(a-3)
$$

If we substitute $x=1$ here again:

$$
4+8-3+a-3=0, \quad a=-6, \quad b=-9
$$

Ede Róka (Ref. g. VI. o. Bp.)

# III. Solution. 

$$
\begin{aligned}
& \left(4 x^{4}+4 x^{3}-11 x^{2}+a x+b\right):\left(x^{2}-2 x+1\right)=4 x^{2}+12 x+9 \\
& \frac{-4 x^{4} \quad \mp 8 x^{3} \pm 4 x^{2}}{12 x^{3}-15 x^{2}}+a x \\
& \frac{-12 x^{3} \mp 24 x^{2} \pm 12 x}{9 x^{2}+(a-12) x}+b
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_b1e1767ab530ccf06fafg-1.jpg?height=103&width=418&top_left_y=1713&top_left_x=932)

The condition for divisibility is that the remainder should vanish for any value of $x$, that is, $a+6=0$ and $b-9=0$, so $a=-6$, $b=9$.

László Lóránd (Bencés $\mathrm{g} . \mathrm{V}$. o. Győr.)