Matematyka

Autorzy:Joanna Czarnowska, Jolanta Wesołowska, Wojciech Babiański, Lech Chańko

Wydawnictwo:Nowa Era

Rok wydania:2014

Rozłóż wielomian w na czynniki liniowe i oblicz jego wartość dla podanego argumentu x 4.33 gwiazdek na podstawie 6 opinii
  1. Liceum
  2. 2 Klasa
  3. Matematyka

Rozłóż wielomian w na czynniki liniowe i oblicz jego wartość dla podanego argumentu x

3
 Zadanie
4
 Zadanie
5
 Zadanie

6
 Zadanie

7
 Zadanie
8
 Zadanie

`a)\ w(x)=(2x-3)(5x-2)+(3-2x)(x+2)=`

`\ \ \ \ \ \ \ \ \ \ =(2x-3)(5x-2)-(2x-3)(x+2)=`

`\ \ \ \ \ \ \ \ \ \ =(2x-3)(5x-2-(x+2))=`

`\ \ \ \ \ \ \ \ \ \ =(2x-3)*(4x-4)`

 

`\ \ \ w(7/2)=(4*7/2-4)*(2*7/2-3)=(14-4)*(7-3)=10*4=40`

 

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

 

 

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

`\ \ \ \ \ \ \ \ \ \ =x(x-5)-(x-5)(3x+1)=`

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

 

`\ \ \ w(6)=(6-5)*(-2*6-1)=1*(-13)=-13`

 

 

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

 

 

 

`c)\ w(x)=(2x^2+x+2)^2-(2x^2-x-4)^2=`

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

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

`\ \ \ \ \ \ \ \ \ \ =(4x+12)(sqrt2x-1)(sqrt2x+1)`

 

`\ \ \ w(-sqrt3/2)=(4*(-sqrt3/2)+12)*(2*(-sqrt3/2)^2-1)=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(-2sqrt3+12)*(2*3/4-1)=(-2sqrt3+12)*1/2=6-sqrt3`

 

 

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

 

 

 

`d)\ w(x)=(x^2-4)^2-(x-2)^2=`

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

 

`\ \ \ \ \ \ \ \ \ \ =(x^2-x-2)*(x^2+x-6)=\ |[Delta_1=(-1)^2-4*1*(-2)=9],[sqrt(Delta_1)=3], [x_1=(1-3)/2=-1], [x_2=(1+3)/2=2],[],[],[Delta_2=1^2-4*1*(-6)=25],[sqrt(Delta_2)=5],[x_3=(-1-5)/2=-3],[x_4=(-1+5)/2=2]|\ =(x+1)(x-2)(x+3)(x-2)=(x+1)(x+3)(x-2)^2`

 

`\ \ \ w(sqrt2-1)=(sqrt2-1+1)*(sqrt2-1+3)*(sqrt2-1-2)^2=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =sqrt2*(sqrt2+2)*(sqrt2-3)^2=(2+2sqrt2)*(2-6sqrt2+9)=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(2+2sqrt2)*(11-6sqrt2)=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =22-12sqrt2+22sqrt2-24=-2+10sqrt2`

 

 

 

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

 

 

 

 

`e)\ w(x)=(9x^2-1)^2-(3x+1)^2=`

`\ \ \ \ \ \ \ \ \ \ =((9x^2-1)-(3x+1))*((9x^2-1)+(3x+1))=`

`\ \ \ \ \ \ \ \ \ \ =(9x^2-3x-2)*(9x^2+3x)=`

`\ \ \ \ \ \ \ \ \ \ =3x(3x+1)(9x^2-3x-2)=\ |[Delta=(-3)^2-4*9*(-2)=9+72=81],[sqrtDelta=9],[x_1=(3-9)/(2*9)=-1/3],[x_2=(3+9)/(2*9)=2/3]|\ =27x(3x+1)(x+1/3)(x-2/3)`

 

 

`\ \ \ w((sqrt3+1)/3)=(9*((sqrt3+1)/3)^2-1)^2-(3*(sqrt3+1)/3+1)^2=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(9*(3+2sqrt3+1)/9-1)^2-(sqrt3+1+1)^2=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(3+2sqrt3)^2-(sqrt3+2)^2=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =9+12sqrt3+12-(3+4sqrt3+4)=`

`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =14+8sqrt3`