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  • Roma
  • Community Manager
2010-03-20T01:17:30+01:00
Dane są wielomiany:
W(x) = (2x + b)( x² + 3x + 1) oraz F(x) = (ax + 3)(x + 1)²
Dla jakich wartości parametrów a,b zachodzi warunek:
W(x) - F(x) = H(x)
gdzie H(x) = x³ + 6x² + 10x + 2?

W(x) - F(x) = (2x + b)( x² + 3x + 1) - [(ax + 3)(x + 1)²] = 2x³ + 6x² + 2x + bx² + 3bx + b - [(ax + 3)(x² + 2x + 1)] = 2x³ + 6x² + 2x + bx² + 3bx + b - (ax³ + 2ax² + ax + 3x² + 6x + 3) = 2x³ + 6x² + 2x + bx² + 3bx + b - ax³ - 2ax² - ax - 3x² - 6x - 3 = 2x³ - ax³ + 3x² + bx² - 2ax² - 4x + 3bx - ax + b - 3 = (2 - a)*x³ + (3 + b - 2a)x² + (- 4 + 3b - a)x + (b - 3)

W(x) - F(x) = H(x)
(2 - a)*x³ + (3 + b - 2a)x² + (- 4 + 3b - a)x + (b + 3) = x³ + 6x² + 10x + 2
stąd
{ 2 - a = 1
{ 3 + b - 2a = 6
{ - 4 + 3b - a = 10
{ b - 3 = 2

2 - a = 1
- a = 1 - 2
- a = - 1 /*(- 1)
a = 1

b - 3 = 2
b = 2 + 3
b = 5

3 + b - 2a = 3 + 5 - 2*1 = 3 + 5 - 2 = 6

- 4 + 3b - a = - 4 + 3*5 - 1 = - 4 + 15 - 1 = 10

{ a = 1
{ b = 5
4 3 4