Toán Lớp 8: Làm tính cộng:
$\frac{x^{2}+x+1}{2x^{3}+4x^{2}+2x}+$ $\frac{x^{2}-x+1}{2x^{3}-2x}+$ $\frac{1}{1+x-x^{2}-x^{3}}$
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Toán Lớp 8: Làm tính cộng:
$\frac{x^{2}+x+1}{2x^{3}+4x^{2}+2x}+$ $\frac{x^{2}-x+1}{2x^{3}-2x}+$ $\frac{1}{1+x-x^{2}-x^{3}}$
Comments ( 2 )
(x^2+x+1)/(2x^3+4x^2+2x)+(x^2-x+1)/(2x^3-2x)+1/(1+x-x^2-x^3)(x>=0;xne+-2)
=(x^2+x+1)/(2x(x^2+2x+1))+(x^2-x+1)/(2x(x^2-1))+1/((1+x)-x^2(x+1))
=(x^2+x+1)/(2x(x+1)^2)+(x^2-x+1)/(2x(x-1)(x+1))-1/((x+1)^2(x-1))
=((x^2+x+1)(x-1))/(2x(x+1)^2(x-1))+((x^2-x+1)(x+1))/(2x(x+1)^2(x-1))-(2x)/(2x(x+1)^2(x-1))
=((x^2+x+1)(x-1)+(x^2-x+1)(x+1)-2x)/(2x(x+1)^2(x-1))
=(x^3-1+x^3+1-2x)/(2x(x+1)^2(x-1))
=(2x^3-2x)/(2x(x+1)^2(x-1))
=(2x(x^2-1))/(2x(x+1)^2(x-1))
=(2x(x-1)(x+1))/(2x(x+1)^2(x-1))
=1/(x+1)
ĐK:x \ne 0,x \ne +-2
\frac{x^2+x+1}{2x^3+4x^2+2x}+\frac{x^2-x+1}{2x^3-2x}+\frac{1}{1+x-x^2-x^3}
=\frac{ x^2+x+1}{2x(x^2+2x+1)}+\frac{x^2-x+1}{2x(x^2-1)}+\frac{1}{(1+x)-x^2(1+x)}
=\frac{x^2+x+1}{2x(x+1)^2}+\frac{x^2-x+1}{2x(x-1)(x+1)}+\frac{1}{(1+x)(1-x^2)}
=\frac{x^2+x+1}{2x(x+1)^2}+\frac{x^2-x+1}{2x(x-1)(x+1)}-\frac{1}{(x+1)^2(x-1)}
=\frac{(x-1)(x^2+x+1)+(x+1)(x^2-x+1)-2x}{2x(x+1)^2(x-1)}
=\frac{x^3-1+x^3+1-2x}{2x(x+1)^2(x-1)}
=\frac{2x^3-2x}{2x(x+1)(x+1)(x-1)}
=\frac{2x(x^2-1)}{2x(x^2-1)(x+1)}
=\frac{1}{x+1}