For what values of the coefficients $A, B, C$ will the lowest power of $x$ in the polynomial form of the following expression be $x^{4}$?

$$
\left(1+A x+B x^{2}+C x^{3}\right)^{3}-(1+x)
$$

In what computational task can we utilize our result?
I. The polynomial form of the given expression is of 9th degree, and it has no constant term because its value is 0 when $x=0$. To satisfy the requirement, the coefficients of $x, x^{2}$, and $x^{3}$ must be 0, that is,

$$
3 A-1=0, \quad 3 A^{2}+3 B=0, \quad A^{3}+6 A B+3 C=0
$$

From the first equation, $A=1 / 3$, substituting this into the second equation gives $B=-A^{2}=-1 / 9$, and substituting both into the third equation gives $C=5 / 81$. However, $x^{4}$ appears, with a coefficient of 10/81. Then the polynomial (1) is

$$
D(x)=\frac{10}{81} x^{4}-\frac{2}{243} x^{5}-\frac{8}{2187} x^{6}+\frac{40}{6561} x^{7}-\frac{25}{19683} x^{8}+\frac{125}{531441} x^{9}
$$

II. In $D(x)$, the sum of the absolute values of the coefficients is $75968 / 5314410, \quad A>1 / 2 \quad$ and

$$
|A-B|=\frac{\left|A^{3}-B^{3}\right|}{A^{2}+A B+B^{2}}<\frac{|\delta(x)|}{A^{2}}<4|\delta(x)|<x^{4}
$$

2. For $x=0.3, -0.3, 0.9$, and $-0.9$, (2) gives the approximate values

$$
1.0917, \quad 0.8883, \quad 1.255, \quad 0.565
$$

compared to the values

$$
1.092, \quad 0.8879, \quad 1.238, \quad 0.465
$$

taken from the table. The last two examples (outside the interval $|x|<1 / 2$) show a larger deviation.