Autorzy:Wojciech Babiański, Lech Chańko, Joanna Czarnowska, Grzegorz Janocha
Wydawnictwo:Nowa Era
Rok wydania:2016
Rozłóż wielomian w na czynniki 4.6 gwiazdek na podstawie 10 opinii

Rozłóż wielomian w na czynniki

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`a)` 

`w(x)=(20x^3-28x^2+8x)(x^4+6x^3+2x^2+12x)=` 

`\ \ \ \ \ \ \ =(4x(5x^2-7x+2))*(x^4+2x^2+6x^3+12x)=`  

`\ \ \ \ \ \ \ =(4x(5x^2-7x+2))*(x^2(x^2+2)+6x(x^2+2))=` 

`\ \ \ \ \ \ \ =(4x(5x^2-7x+2))*(#((x^2+2))^(Delta=0-8<0)(x^2+6x))=`  

`\ \ \ \ \ \ \ =4x#(ul(ul((5x^2-7x+2))))^((**))(x^2+2)(x+6)x=...`  

  

 

`\ \ \ \ \ \ \ \ \ (**)`  

`\ \ \ \ \ \ \ \ \ Delta=(-7)^2-4*5*2=49-40=9` 

`\ \ \ \ \ \ \ \ \ \ sqrtDelta=3` 

`\ \ \ \ \ \ \ \ \ x_1=(7-3)/(2*5)=4/10=2/5` 

`\ \ \ \ \ \ \ \ \ x_2=(7+3)/(2*5)=10/10=1` 

 

 

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

`\ \ \ \ \ \ =20x^2(x-2/5)(x-1)(x+6)(x^2+2)` 

`overline(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )`    

 

 

 

 

 

`b)` 

`w(x)=#((-1/4x^4-2x^3-4x^2))^a#((x^3-7x^2-4x+28))^b=...` 

 

 

`\ \ \ \ \ \ \ a=-1/4x^4-2x^3-4x^2=-1/4x^2(x^2+8x+16)=-1/4x^2(x+4)^2` 

`\ \ \ \ \ \ \ b=x^3-7x^2-4x+28=x^2(x-7)-4(x-7)=(x-7)(x^2-4)=(x-7)(x-2)(x+2)` 

 

 

`...=-1/4x^2(x+4)^2(x-7)(x-2)(x+2)` 

 `overline(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )` 

 

 

 

 

 

`c)` 

`w(x)=#((7x^4+14x^3-21x^2))^a#((x^5-4x^3-x^2+4))^b=...`  

 

 

`\ \ \ \ \ \ \ a=7x^4+14x^3-21x^2=7x^2(x^2+2x-3)=7x^2(x+3)(x-1)`  

`\ \ \ \ \ \ \ \ \ \ \ \ \ Delta=2^2-4*1*(-3)=4+12=16`    

`\ \ \ \ \ \ \ \ \ \ \ \ \ sqrtDelta=4` 

`\ \ \ \ \ \ \ \ \ \ \ \ \ x_1=(-2-4)/2=-6/2=-3` 

`\ \ \ \ \ \ \ \ \ \ \ \ \ x_2=(-2+4)/2=2/2=1` 

 

`\ \ \ \ \ \ \ b=x^5-4x^3-x^2+4=x^3(x^2-4)-1(x^2-4)=(x^2-4)(x^3-1)=`  

`\ \ \ \ \ \ \ \ \ =(x-2)(x+2)(x-1)#((x^2+x+1))^(Delta=1-4<0)` 

 

 

`...=7x^2(x+3)(x-1)(x-2)(x+2)(x-1)(x^2+x+1)=` 

`\ \ \ \ \ =7x^2(x-2)(x-1)^2(x+2)(x+3)(x^2+x+1)` 

`overline(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )` 

 

 

 

 

 

`d)` 

`w(x)=#((3x^4-2x^3+1/3x^2))^a#((x^6-1))^b=...` 

 

`\ \ \ \ \ \ \ \ \ a=3x^4-2x^3+1/3x^2=1/3x^2(9x^2-6x+1)=1/3x^2(3x-1)^2` 

`\ \ \ \ \ \ \ \ \ b=x^6-1=(x^2)^3-1^3=(x^2-1)(x^4+x^2+1)=(x-1)(x+1)(x^4+x^2+1)=` 

`\ \ \ \ \ \ \ \ \ \ \ =(x-1)(x+1)(ul(ul(x^4+2x^2+1))-x^2)=(x-1)(x+1)((x^2+1)^2-x^2)=` 

`\ \ \ \ \ \ \ \ \ \ \ =(x-1)(x+1)#((x^2+1-x))^(Delta=1-4<0)#((x^2+1+x))^(Delta=1-4<0)`  

 

 

`...=1/3x^2(3x-1)^2(x-1)(x+1)(x^2-x+1)(x^2+x+1)`     

`overline(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ )`