Question

wyznacz takie wspolczynniki a,b,c aby W(x)=P(x), gdy W(x)=a(x-2)(x-3) b(x-1)(x-3) +c(x-1)(x-2) i P(x)=5x^2 - 19x + 18 pełne rozwiązanie. na dzis

Answer 1

wyznacz takie wspolczynniki a,b,c aby W(x)=P(x), gdy W(x)=a(x-2)(x-3) b(x-1)(x-3) +c(x-1)(x-2) i P(x)=5x^2 - 19x + 18

 

W(x)=a(x-2)(x-3) b(x-1)(x-3) +c(x-1)(x-2)

W(x) = a(x²-5x +6) ? b(x²-4x +3) +c(x²-3x +2)  

przed b jaki ma być znak  "+" czy "-"

Obliczam dla "+"

 

W(x) = ax² + bx² +cx² -5ax-4bx -3cx  + 6a +3b +2c

W(x) = x²(a +b +c) - x(5a +4b +3c)   + 6a +3b +2c

P(x)  = 5x² - 19x + 18

 

W(x) = P(x) gdy odpowiednie współczynniki przy niewiadomych sa równe

a +b +c = 5

-(5a +4b +3c) = -19      /:(-1)

6a +3b + 2c = 18

 

Powstał układ 3 równań z 3 niewiadomymi

 

a = 5 -b -c

5a +4b +3c = 19

6a +3b + 2c = 18

 

 

a = 5 -b -c

5( 5 -b -c) +4b +3c = 19

6( 5 -b -c) +3b +2c = 18

 

a = 5 -b -c

25 -5b -5c+4b +3c = 19

30 -6b -6c +3b +2c = 18

 

a = 5 -b -c

-b -2c = 19 -25

-3b -4c = 18 -30

 

a = 5 -b -c

-b -2c = -6

-3b -4c = -12

 

a = 5 -b -c

-b = -6 +2c       /:(-1)

-3b -4c = -12

 

a = 5 -b -c

b = 6 -2c

-3(6 -2c) -4c = -12

 

a = 5 -b -c

b = 6 -2c

-18 +6c -4c =  -12

 

a = 5 -b -c

b = 6 -2c

2c = -12 +18

 

a = 5 -b -c

b = 6 -2c

2c =  6       /:2

 

a = 5 -b -c

b = 6 - 2*3 = 6 -6 = 0

c =  3

 

a = 5 -b -c

b = 0

c = 3

 

a = 5 - 0 -3 = 2

b = 0

c = 3

 

a = 2

b =0

c = 3