Thực hiện phép chia. a) (2y^4 - 13y^3 + 15y^2 + 11y - 3) : (y^2 - 4y - 3); b) (5x^3 - 3x^2 + 10) : (x^2 + 1).

Thực hiện phép chia.

a) (2y⁴ - 13y³ + 15y² + 11y - 3) : (y² - 4y - 3);

b) (5x³ - 3x² + 10) : (x² + 1).

a) Thực hiện đặt phép chia ta được:

$\begin{array}{cccccccccc}& 2y^4& -& 13y^3& +& 15y^2& +& 11y& -& 3\\ \overline{}& 2y^4& -& 8y^3& -& 6y^2& & & & \\ & & -& 5y^3& +& 21y^2& +& 11y& -& 3\\ & \overline{}& -& 5y^3& +& 20y^2& +& 15y& & \\ & & & & & y^2& -& 4y& -& 3\\ & & & & \overline{}& y^2& -& 4y& -& 3\\ & & & & & & & & & 0\end{array}\begin{array}{ccccc}{y}^2& -& 4y& -& 3\\ 2y^2& -& 5y& +& 1\\ & & & & \\ & & & & \\ & & & & \\ & & & & \\ & & & & \end{array}$

Vậy (2y⁴ - 13y³ + 15y² + 11y - 3) : (y² - 4y - 3) = 2y² - 5y + 1.

b) Thực hiện phép chia ta được:

$\begin{array}{cccccccc}& 5{\text{x}}^3& -& 3{\text{x}}^2& & & +& 10\\ \overline{}& 5{\text{x}}^3& & & +& 5\text{x}& & \\ & & -& 3{\text{x}}^2& -& 5x& +& 10\\ & \overline{}& -& 3{\text{x}}^2& & & -& 3\\ & & & & -& 5\text{x}& +& 13\end{array}\begin{array}{ccc}{x}^2& +& 1\\ 5\text{x}& -& 3\\ & & \\ & & \\ & & \end{array}$

Vậy 5x³ - 3x² + 10 = (5x - 3)(x² + 1) + (-5x + 13) = (5x - 3)(x² + 1) - 5x + 13.