Question

Rozłóż wielomian w, korzystając z trójmianu kwadratowego na czynniki

1. w(x)= 2x^5 - 8x^4

2. w(x)= -1/3x^5 + 8/3x^4 + 3x^3

3. w(x)= x^2 + √2x^2 + x

4. w(x)= (2x^2 - 5x - 3)(x^3 + 2x^2 - 15x)

Answer 1

z.1

W(x) = 2 x^5 - 8 x^4 = 2 x^4 *( x - 4)

z.2

W(x) = ( -1/3) x^5 + (8/3) x^4  + 3 x^3 =  -  x^3 *(   (1/3) x^2 - ( 8/3) x - 3) = 

= ( - 1/3) x^3 *( x^2 - 8 x - 9 ) = ( - 1/3) x^3 *( x - 9)*( x + 1)

z.3

W(x) =  ?

 

z.4

W(x) = ( 2 x^2 - 5 x - 3)*( x^3 + 2 x^2 - 15 x) =

 

delta 1 = ( -5)^2 - 4*2*(-3) = 25 + 24 = 49

p ( delta1) = 7

x1 = ( 5 - 7)/ 4 = - 1/2

x2 = ( 5 + 7) / 4 = 3

więc

2 x^2 - 5 x - 3 = 2*( x + 0,5 )*( x - 3)

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x^3 + 2 x^2 - 15 x = x*( x^2 + 2 x - 15 )

delta 2 = 2^2 - 4*1*( -15) = 4 + 60 = 64

p ( delta 2) = 8

x3 = ( - 2 - 8) / 2 =- 5

x4 = ( - 2 + 8) / 2 = 3

więc

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

zatem

W(x) = 2*( x + 0,5)*( x - 3)* x *( x + 5)*( x - 3)

lub

W(x) = 2x*( x + 0,5)*( x + 5)*( x - 3)^2

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