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Giải bài 2.21 trang 50 SGK Toán 8 - Cùng khám phá

Rút gọn biểu thức sau theo hai cách
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Đề bài

Rút gọn biểu thức sau theo hai cách (sử dụng và không sử dụng tính chất phân phối của phép nhân đối với phép cộng): \(\frac{{x + 1}}{x}.\left( {{x^2} - x + 1 - \frac{{{x^2}}}{{x + 1}}} \right)\).

Phương pháp giải - Xem chi tiết

Sử dụng và không sử dụng tính chất phân phối của phép nhân đối với phép cộng để rút gọn biểu thức thành hai cách.

Lời giải chi tiết

Cách 1:\(\begin{array}{l}\frac{{x + 1}}{x}.\left( {{x^2} - x + 1 - \frac{{{x^2}}}{{x + 1}}} \right)\\ = \frac{{x + 1}}{x}.\left( {\frac{{{x^2}\left( {x + 1} \right)}}{{x + 1}} - \frac{{x\left( {x + 1} \right)}}{{x + 1}} + \frac{{x + 1}}{{x + 1}} - \frac{{{x^2}}}{{x + 1}}} \right)\\ = \frac{{x + 1}}{x}.\left( {\frac{{{x^3} + {x^2} - {x^2} - x + x + 1 - {x^2}}}{{x + 1}}} \right)\\ = \frac{{x + 1}}{x}.\left( {\frac{{{x^3} - {x^2} + 1}}{{x + 1}}} \right) = \frac{{\left( {x + 1} \right)\left( {{x^3} - {x^2} + 1} \right)}}{{x.\left( {x + 1} \right)}} = \frac{{{x^3} - {x^2} + 1}}{x}\end{array}\)Cách 2:\(\begin{array}{l}\frac{{x + 1}}{x}.\left( {{x^2} - x + 1 - \frac{{{x^2}}}{{x + 1}}} \right)\\ = \frac{{x + 1}}{x}.{x^2} - \frac{{x + 1}}{x}.x + \frac{{x + 1}}{x} - \frac{{x + 1}}{x}.\frac{{{x^2}}}{{x + 1}}\\ = x\left( {x + 1} \right) - \left( {x + 1} \right) + \frac{{x + 1}}{x} - x\\ = {x^2} + x - x - 1 + \frac{{x + 1}}{x} - x\\ = {x^2} - 1 - x + \frac{{x + 1}}{x}\\ = \frac{{{x^3} - x - {x^2} + x + 1}}{x} = \frac{{{x^3} - {x^2} + 1}}{x}\end{array}\)

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