Tính:

a) (6x² - 2x + 1) : (3x - 1);

b) (27x³ + x² - x + 1) : (-2x + 1);

c) (8x³ + 2x² + x) : (2x³ + x + 1);

d) (3x⁴ + 8x³ - 2x² + x + 1) : (3x + 1).

a) Thực hiện phép tính ta được:

$\begin{array}{cccccc}& 6{\text{x}}^2& -& 2\text{x}& +& 1\\ \overline{}& 6{\text{x}}^2& -& 2\text{x}& & \\ & & & & & 1\end{array}\begin{array}{ccc}3\text{x}& -& 1\\ 2\text{x}& & \\ & & \end{array}$

Vậy 6x2 - 2x + 1 = 2x . (3x - 1) + 1.

b) Thực hiện phép tính ta được:

$\begin{array}{cccccc}& 27{\text{x}}^3+x^2-x+1& & & & \\ \overline{}& 27{\text{x}}^3-\frac{27}{2}{x}^2& & & & \\ & \frac{29}{2}{x}^2-x+1& & & & \\ \overline{}& \frac{29}{2}{x}^2-\frac{29}{4}x& & & & \\ & \frac{25}{4}x+1& & & & \\ \overline{}& \frac{25}{4}x-\frac{25}{8}& & & & \\ & \frac{33}{8}& & & & \end{array}\begin{array}{ccc}-2\text{x}+1& & \\ \frac{-27}{2}{x}^2-\frac{29}{4}x-\frac{25}{8}& & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \end{array}$

Vậy 27x3 + x2 - x + 1 = $\left(\frac{-27}{2}{x}^2-\frac{29}{4}x-\frac{25}{8}\right)$ . (-2x + 1) + $\frac{33}{8}$.

c) Thực hiện phép tính ta được:

$\begin{array}{cccccccc}& 8{\text{x}}^3& +& 2{\text{x}}^2& +& x& & \\ \overline{}& 8{\text{x}}^3& & & +& 4\text{x}& +& 4\\ & & & 2x^2& -& 3\text{x}& -& 4\end{array}\begin{array}{ccccc}2{\text{x}}^3& +& x& +& 1\\ 4& & & & \\ & & & & \end{array}$

Vậy 8x3 + 2x2 + x = 4(2x3 + x + 1) + (-3x - 4).

d) Thực hiện phép tính ta được:

$\begin{array}{cc}& 3{\text{x}}^4+8{\text{x}}^3-2{\text{x}}^2+x+1\\ \overline{}& 3{\text{x}}^4+x^3\\ & 7{\text{x}}^3-2{\text{x}}^2+x+1\\ \overline{}& 7{\text{x}}^3+\frac{7}{3}{x}^2\\ & \frac{-13}{3}{x}^2+x+1\\ \overline{}& \frac{-13}{3}{x}^2-\frac{13}{9}x\\ & \frac{22}{9}x+1\\ \overline{}& \frac{22}{9}x+\frac{22}{27}\\ & \frac{5}{27}\end{array}\begin{array}{c}3\text{x}+1\\ x^3+\frac{7}{3}{x}^2-\frac{13}{9}x+\frac{22}{27}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{array}$