Rozłóż wielomian na czynniki;a) W(x)=(x-1) (x²-3)-(x+3) (x-1)b) W(x)=(2x-3) (X²-2x)-(2x-3)²c) W(x)= .... Question from @konrad57

wik8947201

W(x)=(x-1)(x²-3-x-3)=(x-1)(x²-x-6)=(x-1)(x²+2x-3x-6)=(x-1)[x(x+2)-3(x+2)]=

=(x-1)(x+2)(x-3)

W(x)=x(x-2)(2x-3)-(2x-3)²=(2x-3)[x(x-2)-(2x-3)]=(2x-3)(x²-2x-2x+3)=(2x-3)(x²-4x+3)=

(2x-3)(x²-x-3x+3)=(2x-3)[x(x-1)-3(x-1)]=(2x-3)(x-1)(x-3)

W(x)=(x-3)(x+3)(x+1)-x(x-3)(2x+4)=(x-3)(x²+4x+3-2x²-4x)=(x-3)(-x²+3)=

(x-3)(√3-x)(√3+x)

W(x,y)=4-(25x²+10xy+y²)=4-(5x+y)²=(2-5x-y)(2+5x+y)

W(x,y)=(3x-1)²-25y²=(3x-5y-1)(3x+5y-1)

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Damato

$\\\\ a) \\\\ W(x)=(x-1)(x^2-3-x-3)=(x-1)(x^2-x-6)=\\\\ = (x-1)(x+2)(x-3)\\\\ x^2-x-6=0 \\\\\\ a=1,b=-1,c=-6 \\\\ \Delta=b^2-4ac=(-1)^2-4 \cdot 1 \cdot (-6)=1+24=15 \\\\ \sqrt{\Delta}=\sqrt{25}=5 \\\\ x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{1-5}{2}=\frac{-4}{2}=-2 \\\\ x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{1+5}{2}=\frac{6}{2}=3 \\\\ Postac \ \ iloczynowa: \\\\ a(x-x_1)(x-x_2) \ \ czyli: \\\\ x^2-x-6=(x+2)(x-3)$

$\\\\ b) \\\\ W(x)=(2x-3) (x^2-2x)-(2x-3)^2=\\\\ (x-2)(2x-3)-(2x-3)^2=\\\\ =(2x-3)[x(x-2)-(2x-3)]=\\\\ =(2x-3)(x^2-2x-2x+3)=(2x-3)(x^2-4x+3)= \\\\ = (2x-3)(x^2-x-3x+3)=\\\\ = (2x-3)[x(x-1)-3(x-1)]=(2x-3)(x-1)(x-3)$

$\\\\ c) W(x)=(x^2-9)(x+1)-(x^2-3x)(2x+4)=\\\\ = (x-3)(x+3)(x+1)-x(x-3)(2x+4)=\\\\ =(x-3)(x^2+4x+3-2x^2-4x)=(x-3)(-x^2+3)=\\\\ = (x-3)(\sqrt{3}-x)(\sqrt{3}+x) \\\\\\ d) \\\\ W(x)=4-25x^2-10xy-y^2=4-(25x^2+10xy+y^2)=\\\\ = 4-(5x+y)^2== 4-(5x+y)(5x-y)=\\\\ =(2-5x-y)(2+5x+y) \\\\\\ e) \\\\ W(x)=9x^2-6x+1-25y^2=(3x-1)^2-25y^2=\\\\ =(3x-1)(3x+1)-25y^2=\\\\ =(3x-5y-1)(3x+5y-1)$

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