Rozłóż na czynniki wielomiany:a) b) c).... Question from @werson37

werson37 @werson37

November 2018 2 7 Report

Rozłóż na czynniki wielomiany:

a) $w(x)=x^{4}+10x^{3}-90x-81$

b) $G(x)=x^{4}-625$

c) $H(x)=(x^{2}+4x-1)^{2}-16$

krysta

$W(x)=x^4+10x^3-90x-81=(x^4-81)+(10x^3-81)=(x^2-9)(x^2+9)+10x(x^2-9)=(x^2-9)(x^2+10x+9)= (x-3)(x+3)(x^2+9x+x+9)= (x-3)(x+3)(x(x+9)+(x+9))= (x-3)(x+3)((x+9)(x+1)) = (x-3)(x+3)(x+9)(x+1)$ )

$G(x)=x^4-625 = (x^2-25)(x^2+25)= (x-5)(x+5)(x^2+25)$

$H(x)=(x^2+4x-1)^2-16 = (x^2+4x-1+4)(x^2+4x-1-4)=(x^2+4x+3)(x^2+4x-5)\\x^2+4x+3 = 0\\\Delta =b^2-4ac=4^2-4\cdot 1\cdot 3=16-12=4\\\sqrt{\Delta }=\sqrt{4}=2\\x_{1}=\frac{-b-\sqrt{\Delta }}{2a} =\frac{-4-2}{2}=\frac{-6}{2}=-3\\x_{2}=\frac{-b+\sqrt{\Delta }}{2a} =\frac{-4+2}{2}=\frac{-2}{2}=-1$

$x^2+4x-5=0\\\sqrt{\Delta }=b^2-4ac = 4^2-4\cdot 1\cdot (-5)=16+20=36\\\sqrt{\Delta =}\sqrt{36}=6\\x_{1}=\frac{-b-\sqrt{\Delta }}{2a} =\frac{-4-6}{2}=\frac{-10}{2}=-5\\x_{2}=\frac{-b+\sqrt{\Delta }}{2a} =\frac{-4+6}{2}=\frac{2}{2}=1\\H(x)=(x^2+4x-1)^2-16 = (x^2+4x-1+4)(x^2+4x-1-4)=(x^2+4x+3)(x^2+4x-5)= (x+3)(x+1)(x+5)(x-1)$

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mutopompka

$> B) <img src=$

$H(x)=(x^{2}+4x-1)^{2}-16\\ H(x)=[(x^2+4x-1)-4][(x^2+4x-1)+4]\\ H(x)=(x^2+4x-5)(x^2+4x+3)\\ H(x)=(x^2+5x-x-5)(x^2+3x+x+3)\\ H(x)=(x^2-x+5(x-1))(x^2+x+3(x+1))\\ H(x)=[x(x-1)+5(x-1)][x(x+1)+3(x+1)]\\ H(x)=[(x-1)(x+5][(x+1)(x+3)]\\ \\ H(x)=(x-1)(x+1)(x+3)(x+5)$

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