Question

Wielomiany W(x) = a(x-2)(x-3) + b(x-1)(x-3) + c(x-1)(x-2) i G(x) = 5x^2 - 19x + 18 są równe. Wyznacz a, b, c.

Answer 1

W(x)=a(x²-5x+6)+b(x²-4x+3)+c(x²-3x+2)

W(x)=(a+b+c)x²-(5a+4b+3c)x+(6a+3b+2c)

a+b+c=5⇒a=5-b-c

5a+4b+3c=19

6a+3b+2c=18

 

5(5-b-c)+4b+3c=19

6(5-b-c)+3b+2c=18

 

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

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

 

-b-2c=-6  /*(-2)

-3b-4c=-12

 

2b+4c=12

-3b-4c=-12

--------------- +

-b=0

b=0

-2c=-6

c=3

a=5-3=2

 

Odp. a=2, b=0, c=3

 

spr.

W(x)=2(x-2)(x-3)+3(x-1)(x-2)=(x-2)(2x-6+3x-3)=(x-2)(5x-9)=5x²-19x+18