Записать формулу деления многочлена p(x) на многочлен Q(x)

1) P(x) = x² + 3x + 4; Q(x) = x – 2;

P(x) = x² - 4x + 4 + 4x - 4 + 3x + 4 = (x-2)² + 7x = (x-2)² + 7(x - 2) + 14 =

= (x - 2) * (x - 2 + 7) + 14 = (x - 2) * (x + 5) + 14.

3) P(x) = 6x³ + 3x² - 4x + 3; Q(x) = 2x + 1

P(x) = 6/8 * (8x³ + 3 * 4x² + 3 * 2x + 1) - 6/8 * 12x² - 6/8 * 6x - 6/8 + 3x² - 4x + 3

= ¾ * (2x+1)³ - 9x² - 9/2 * x - 3/4 + 3x² - 4x + 3 =

= ¾ * (2x+1)³ - 6x² - 17/2 * x + 9/4 =

= ¾ * (2x+1)³ - 3/2 * (4x² + 4x + 1) + 6x + 3/2 - 17/2 * x + 9/4 =

= ¾ * (2x+1)³ - 3/2 * (2x+1)² - 5/2 * x + 15/4 =

= ¾ * (2x+1)³ - 3/2 * (2x+1)² - 5/4 * (2x + 1) + 5/4 + 15/4 =

= ¾ * (2x+1)³ - 3/2 * (2x+1)² - 5/4 * (2x + 1) + 5 =

= ¼ * (2x + 1) * (2(2x + 1)² - 6(2x + 1) - 5) + 5.

Можно упростить во второй скобке.

4) P(x) = 2x³ - 3x² + 2x - 2; Q(x) = x² + 2.

P(x) = 2(x³ + 2x) - x - 3x² + 2x - 2 =

= 2x * (x² + 2) - 3x² - 2x - 2 = 2x * (x² + 2) - 3(x² + 2) - 2x - 2 + 6 =

= (x² + 2) * (2x - 3) - 2x + 4.