rozłóż na czynniki:
\(\displaystyle{ 1: \ (ab+ac+bc)(a+b+c)-abc\\
2: \ (a+b+c)^{3}-a^{3}-b^{3}-c^{3} \\
3: \ a^{3}+b^{3}+c^{3} -3abc \\
4: \ y^{3}(z-x)-x^{3}(z-y)+z^{3}(x-y)}\)
Rozkładanie wielomianów na czynniki
- RyHoO16
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Rozkładanie wielomianów na czynniki
1.
\(\displaystyle{ (a+b+c)(ab+bc+ac)-abc=a^2b+a^2c+ac^2+b^2a+b^2c+bc^2+2abc= \\
a(ab+ac+bc+c^2)+b(ab+ac+bc+c^2)=(a+b)(a(b+c)+c(b+c))=(a+b)(b+c)(a+c)}\)
\(\displaystyle{ (a+b+c)(ab+bc+ac)-abc=a^2b+a^2c+ac^2+b^2a+b^2c+bc^2+2abc= \\
a(ab+ac+bc+c^2)+b(ab+ac+bc+c^2)=(a+b)(a(b+c)+c(b+c))=(a+b)(b+c)(a+c)}\)
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Rozkładanie wielomianów na czynniki
2.
\(\displaystyle{ (a+b+c)^{3} = a^{3} + b^{3} +c^{3} + 3a^{2}b+3a^{2}c+3ab^{2}+3b^{2}c+3ac^{2}+3bc^{2}+6abc}\)
\(\displaystyle{ (a+b+c)^{3} - a^{3}-b^{3}-c^{3}= a^{3} + b^{3} +c^{3} + 3a^{2}b+3a^{2}c+3ab^{2}+3b^{2}c+3ac^{2}+3bc^{2}+6abc - a^{3}-b^{3}-c^{3}=3a^{2}b+3a^{2}c+3ab^{2}+3b^{2}c+3ac^{2}+3bc^{2}+6abc = 3[a^{2}b+a^{2}c+ab^{2}+b^{2}c+ac^{2}+bc^{2}+2abc]=3[a(ab+ac+bc+c^{2})+b(ab+ac+bc
+c^{2})]=3[(a+b)(a(b+c)+c(b+c))]=3(a+b)(b+c)(a+c)}\)
\(\displaystyle{ (a+b+c)^{3} = a^{3} + b^{3} +c^{3} + 3a^{2}b+3a^{2}c+3ab^{2}+3b^{2}c+3ac^{2}+3bc^{2}+6abc}\)
\(\displaystyle{ (a+b+c)^{3} - a^{3}-b^{3}-c^{3}= a^{3} + b^{3} +c^{3} + 3a^{2}b+3a^{2}c+3ab^{2}+3b^{2}c+3ac^{2}+3bc^{2}+6abc - a^{3}-b^{3}-c^{3}=3a^{2}b+3a^{2}c+3ab^{2}+3b^{2}c+3ac^{2}+3bc^{2}+6abc = 3[a^{2}b+a^{2}c+ab^{2}+b^{2}c+ac^{2}+bc^{2}+2abc]=3[a(ab+ac+bc+c^{2})+b(ab+ac+bc
+c^{2})]=3[(a+b)(a(b+c)+c(b+c))]=3(a+b)(b+c)(a+c)}\)
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Rozkładanie wielomianów na czynniki
4) \(\displaystyle{ y^{3}(z-x)-x^{3}(z-y)+z^{3}(x-y)=y^{3}z-y^{3}x-x^{3}z+x^{3}y+z^{3}x-z^{3}y=}\)
\(\displaystyle{ =y(y^{2}z-y^{2}x+x^{3}-z^{3})-xz(x^{2}-z^{2})=}\)
\(\displaystyle{ =y[y^{2}(z-x)+(x-z)(x^{2}+xz+z^{2})]-xz(x-z)(x+z)=}\)
\(\displaystyle{ =y[-y^{2}(x-z)+(x-z)(x^{2}+xz+z^{2})]-xz(x-z)(x+z)=}\)
\(\displaystyle{ =y[(x-z)(-y^{2}+x^{2}+xz+z^{2})]-xz(x-z)(x+z)=}\)
\(\displaystyle{ =(x-z)[y(-y^{2}+x^{2}+xz+z^{2})-xz(x+z)]=}\)
\(\displaystyle{ =(x-z)[y \left( (x-y)(x+y)+z(x+z) \right)-xz(x+z)]=}\)
\(\displaystyle{ =(x-z)[y(x-y)(x+y)+zy(x+z)-xz(x+z)]=}\)
\(\displaystyle{ =(x-z)[y(x-y)(x+y)+(x+z)(zy-xz)]=}\)
\(\displaystyle{ =(x-z)[y(x-y)(x+y)-z(x+z)(x-y)]=}\)
\(\displaystyle{ =(x-z)[(x-y) \left( (xy+y^{2}-z(x+z) \right)]=(x-z)(x-y)(xy+y^{2}-zx-z^{2})=}\)
\(\displaystyle{ =(x-z)(x-y)[x(y-z)+(y-z)(y+z)]=(x-z)(x-y)[(y-z)(x+y+z)]=}\)
\(\displaystyle{ =(x-z)(x-y)(y-z)(x+y+z)}\)
\(\displaystyle{ =y(y^{2}z-y^{2}x+x^{3}-z^{3})-xz(x^{2}-z^{2})=}\)
\(\displaystyle{ =y[y^{2}(z-x)+(x-z)(x^{2}+xz+z^{2})]-xz(x-z)(x+z)=}\)
\(\displaystyle{ =y[-y^{2}(x-z)+(x-z)(x^{2}+xz+z^{2})]-xz(x-z)(x+z)=}\)
\(\displaystyle{ =y[(x-z)(-y^{2}+x^{2}+xz+z^{2})]-xz(x-z)(x+z)=}\)
\(\displaystyle{ =(x-z)[y(-y^{2}+x^{2}+xz+z^{2})-xz(x+z)]=}\)
\(\displaystyle{ =(x-z)[y \left( (x-y)(x+y)+z(x+z) \right)-xz(x+z)]=}\)
\(\displaystyle{ =(x-z)[y(x-y)(x+y)+zy(x+z)-xz(x+z)]=}\)
\(\displaystyle{ =(x-z)[y(x-y)(x+y)+(x+z)(zy-xz)]=}\)
\(\displaystyle{ =(x-z)[y(x-y)(x+y)-z(x+z)(x-y)]=}\)
\(\displaystyle{ =(x-z)[(x-y) \left( (xy+y^{2}-z(x+z) \right)]=(x-z)(x-y)(xy+y^{2}-zx-z^{2})=}\)
\(\displaystyle{ =(x-z)(x-y)[x(y-z)+(y-z)(y+z)]=(x-z)(x-y)[(y-z)(x+y+z)]=}\)
\(\displaystyle{ =(x-z)(x-y)(y-z)(x+y+z)}\)
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Rozkładanie wielomianów na czynniki
ad. 3)
\(\displaystyle{ x^{3}+y^{3}+z^{3}-3xyz= \\ =(x+y+z)(x^{2}+y^{2}+z^{2})-3xyz-xy^{2}-xz^{2}-yx^{2}-yz^{2}-zx^{2}-zy^{2}= \\ =
(x+y+z)(x^{2}+y^{2}+z^{2})-xy(z+y+x)-yz(x+z+y)-zx(y+z+x)=(x+y+z)(x^{2}+y^{2}+z^{2})-(x+y+z)(xy+yz+zx)= \\ =
(x+y+z)(x^{2}+y^{2}+z^{2}-xy-yz-zx)=\frac{1}{2}(x+y+z)((x-y)^{2}+(y-z)^{2}+(z-x)^{2})}\)
\(\displaystyle{ x^{3}+y^{3}+z^{3}-3xyz= \\ =(x+y+z)(x^{2}+y^{2}+z^{2})-3xyz-xy^{2}-xz^{2}-yx^{2}-yz^{2}-zx^{2}-zy^{2}= \\ =
(x+y+z)(x^{2}+y^{2}+z^{2})-xy(z+y+x)-yz(x+z+y)-zx(y+z+x)=(x+y+z)(x^{2}+y^{2}+z^{2})-(x+y+z)(xy+yz+zx)= \\ =
(x+y+z)(x^{2}+y^{2}+z^{2}-xy-yz-zx)=\frac{1}{2}(x+y+z)((x-y)^{2}+(y-z)^{2}+(z-x)^{2})}\)