Tính:

a) (x³ + 1) : (x² - x + 1);

b) (8x³ - 6x² + 5) : (x² + x + 1).

a) Thực hiện phép tính ta được:

$\begin{array}{cccccc}& x^2& -& 2\text{x}& +& 1\\ \overline{}& x^2& -& x& & \\ & & -& x& +& 1\\ & \overline{}& -& x& +& 1\\ & & & & & 0\end{array}\begin{array}{ccc}x& -& 1\\ x& -& 1\\ & & \\ & & \\ & & \end{array}$

Vậy (x2 - 2x + 1) : (x - 1) = x - 1.

b) Thực hiện phép tính ta được:

$\begin{array}{cccccc}& x^3& +& 2{\text{x}}^2& +& x\\ \overline{}& x^3& +& x^2& & \\ & & & x^2& +& x\\ & & \overline{}& x^2& +& x\\ & & & & & 0\end{array}\begin{array}{ccc}{x}^2& +& x\\ x& +& 1\\ & & \\ & & \\ & & \end{array}$

Vậy (x3 + 2x2 + x) : (x2 + x) = x + 1.

c) Thực hiện phép tính ta được:

$\begin{array}{cccccc}& -16{\text{x}}^4& & & +& 1\\ \overline{}& -16{\text{x}}^4& +& 4{\text{x}}^2& & \\ & & & -4{\text{x}}^2& +& 1\\ & & \overline{}& -4{\text{x}}^2& +& 1\\ & & & & & 0\end{array}\begin{array}{ccc}-4{\text{x}}^2& +& 1\\ 4{\text{x}}^2& +& 1\\ & & \\ & & \\ & & \end{array}$

Vậy (-16x4 + 1) : (-4x2 + 1) = 4x2 + 1.

d) Thực hiện phép tính ta được:

$\begin{array}{cccccccccccc}& -32{\text{x}}^5& & & & & & & & & +& 1\\ \overline{}& -32{\text{x}}^5& +& 16{\text{x}}^4& & & & & & & & \\ & & & -16{\text{x}}^4& & & & & & & +& 1\\ & & \overline{}& -16{\text{x}}^4& +& 8{\text{x}}^3& & & & & & \\ & & & & & -8{\text{x}}^3& & & & & +& 1\\ & & & & \overline{}& -8{\text{x}}^3& +& 4{\text{x}}^2& & & & \\ & & & & & & & -4{\text{x}}^2& & & +& 1\\ & & & & & & \overline{}& -4{\text{x}}^2& +& 2\text{x}& & \\ & & & & & & & & & -2\text{x}& +& 1\\ & & & & & & & & \overline{}& -2\text{x}& +& 1\\ & & & & & & & & & & & 0\end{array}\begin{array}{ccccccccc}-2\text{x}& +& 1& & & & & & \\ 16{\text{x}}^4& +& 8{\text{x}}^3& +& 4{\text{x}}^2& +& 2\text{x}& +& 1\\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \\ & & & & & & & & \end{array}$

Vậy (-32x5 + 1) : (-2x + 1) = 16x4 + 8x3 + 4x2 + 2x + 1.

a) (8x³ + 2x² - 6x) : (4x)

= (8x³ : 4x) + (2x² : 4x) - (6x : 4x)

= 2x² + $\frac{1}{2}$x - $\frac{3}{2}$.

b) (5x³ - 4x) : (-2x)

= [5x³ : (-2x)] - [4x : (-2x)]

= $\frac{-5}{2}$x² - (-2)

= $\frac{-5}{2}$x² + 2.

c) (-15x⁶ - 24x³) : (-3x²)

= (-15x⁶) : (-3x²) - 24x³ : (-3x²)

= 5x⁴ - (-8x)

= 5x⁴ + 8x.