Напишите формулу деления многочлена P (x) на многочлен Q (x) ? 1) P (x) = x2+3x+4 Q (x) = x-2 3) P (x) = 6x3+3x2-4x+3 Q (x) = 2x+1 4) Р (х) = 2 х3-3 х2+2 х-2 Q (x) = x2+2

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Математика

Матвей

19 сентября, 01:09

Напишите формулу деления многочлена P (x) на многочлен Q (x) ? 1) P (x) = x2+3x+4 Q (x) = x-2 3) P (x) = 6x3+3x2-4x+3 Q (x) = 2x+1 4) Р (х) = 2 х3-3 х2+2 х-2 Q (x) = x2+2

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Киля · Answer 1

1) P (x) = x^2 + 3x + 4; Q (x) = x - 2

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

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

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

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

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

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

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

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

= 1/4 * (2x + 1) * (2 (2x+1) ^2 - 6 (2x+1) - 5) + 5

Можно упростить во второй скобке.

4) P (x) = 2x^3 - 3x^2 + 2x - 2; Q (x) = x^2 + 2

P (x) = 2 (x^3 + 2x) - 4x - 3x^2 + 2x - 2 =

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

= (x^2 + 2) * (2x - 3) - 2x + 4